This can help you to develop a deeper understanding of math and how it applies to the real world, which can be useful in a variety of fields such as science, engineering, and finance. After this lesson you'll be able to simplify expressions with exponents. This is amazing, it helped me so often already! If you're looking for a tutor who can help you with any subject, look no further than Instant Expert Tutoring. Related Symbolab blog posts . As, in India, schools are closed so this is a very helpful app also for learning and answering for anyone, at first, when I took pictures with the camera it didn't always work, I didn't receive the answer I was looking for, because this app is so useful and easily accessable, my teacher doesn't allow it but they don't know that it shows you how to solve the problem which I think is awesome. Look at the image given below showing another simplifying expression example. Indulging in rote learning, you are likely to forget concepts. To simplify algebraic expressions, follow the steps given below: Let us take an example for a better understanding. Simplify expressions with positive exponents calculator - This Simplify expressions with positive exponents calculator helps to fast and easily solve any math. . This tool is designed to take the frustration out of algebra by helping you to simplify and reduce your expressions to their simplest form. Let us take one more example to understand it. There are several steps you can follow to simplify an algebraic expression: Combine like terms: The first step in simplifying an expression is to look for terms with the same variables and exponents and combine them using the appropriate operations. Step 1: Enter the expression you want to simplify into the editor. Simplify, Simplify (a12b)12(ab12) . How to simplify expressions with exponents calculator - Simplifies expressions step-by-step and shows the work! Next, x^2 divided by x^4 is x^(2-4). Recall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the expression more simply with fewer terms. I highly recommend you use this site! We're almost done: 2 times p^(1-3) is -2, times q^(2-4), which is q^(-2) times r^9. We distribute the exponent to everything in the parenthesis. This appears later in more advanced courses, but for now, we will consider the value to be undefined. Write each of the following quotients with a single base. To simplify an algebraic expression means to rewrite it in a simpler form, without changing its value. Simplifying Exponents. Solve Now How to Simplify Exponents or Powers on the TI When using the power rule, a term in exponential notation is raised to a power. Therefore, 3/4x + y/2 (4x + 7) = 3/4x + 2xy + 7y/2. Simplify In a similar way to the product rule, we can simplify an expression such as \displaystyle \frac { {y}^ {m}} { {y}^ {n}} ynym, where \displaystyle m>n m > n. A fully demonstrated steps by steps solution of a numerical (not a question), awesome makes life easy and has saved me an enormous amount of time the app is worth 20 dollars a month. On most calculators, you enter the base, press the exponent key and enter the exponent. In this equation, you'd start by simplifying the part of the expression in parentheses: 24 - 20. This is our simplified answer with positive exponents. 986+ Experts. Using a calculator, we enter [latex]2,048\times 1,536\times 48\times 24\times 3,600[/latex] and press ENTER. Choose "Simplify" from the topic selector and click to see the result in our Algebra Calculator! Also, the product and quotient rules and all of the rules we will look at soon hold for any integer [latex]n[/latex]. Analytical geometry of two and three dimensions in hindi, How do you subtract fractions step by step, How to find the volume of a prism with fractions, How to improve function of pituitary gland, Math problem solving worksheets for grade 1, What do vampires do on halloween math worksheet answers, What is the order of differential equation given by dy/dx+4y=sinx. Introduction Exponents can be attached to variables as well as numbers. Simplify the expression \frac { { { {x}^ {2}}}} { { { {x}^ { {-3}}}}} x3x2. Explore the use of several properties used to simplify expressions with exponents, including the product of powers, power to a power, quotient of powers, power of a product, and the zero property. For instance, consider [latex]{\left(pq\right)}^{3}[/latex]. To simplify an expression with fractions find a common denominator and then combine the numerators. Solve an equation, inequality or a system. Write each of the following products with a single base. Yes. This implies, 2ab + 4b (b2) - 4b (2a). Write answers with positive exponents. Use exponent rules to simplify terms with exponents, if any. Welcome to our step-by-step math solver! One way to think about math equations is to think of them as a puzzle. Flash cards are a fantastic and easy way to memorize topics, especially math properties. Enter an exponential expression below which you want to simplify. It can be very useful while simplifying expressions. Simplifying expressions with exponents In the term , is the base and is the exponent. It includes four examples. Used with the function expand, the function simplify can expand and collapse a literal expression. Example of Dividing Monomials When you divide monomial expressions, subtract the exponents of like bases. The power of a quotient of factors is the same as the quotient of the powers of the same factors. ti 89 algebra discovery distributive property nc discrete math practice problems rational expressions calculator using excel to find least common number from [latex]\begin{array}{ccc}\hfill {\left({x}^{2}\right)}^{3}& =& \stackrel{{3\text{ factors}}}{{{\left({x}^{2}\right)\cdot \left({x}^{2}\right)\cdot \left({x}^{2}\right)}}}\hfill \\ & =& \hfill \stackrel{{3\text{ factors}}}{{{\left(\stackrel{{2\text{ factors}}}{{\overbrace{x\cdot x}}}\right)\cdot \left(\stackrel{{2\text{ factors}}}{{\overbrace{x\cdot x}}}\right)\cdot \left(\stackrel{{2\text{ factors}}}{{\overbrace{x\cdot x}}}\right)}}}\\ & =& x\cdot x\cdot x\cdot x\cdot x\cdot x\hfill \\ & =& {x}^{6}\hfill \end{array}[/latex], [latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex]. We start at the beginning. You can also use the calculator to check your work and ensure that you have correctly simplified your expression. Follow the PEMDAS rule to determine the order of terms to be simplified in an expression. BYJU'S online negative exponents calculator tool makes the calculation faster, and it displays the result in a fraction of seconds. Distributive property states that an expression given in the form of x (y + z) can be simplified as xy + xz. Plus, get practice tests, quizzes, and personalized coaching to help you The equations section lets you solve an equation or system of equations. Simplify algebraic expressions with exponents. Simplify There's one exponent in this equation: 42, or four to the second power. For any real number [latex]a[/latex] and natural numbers [latex]m[/latex] and [latex]n[/latex], the product rule of exponents states that. [latex]{x}^{2}\cdot {x}^{5}\cdot {x}^{3}=\left({x}^{2}\cdot {x}^{5}\right)\cdot {x}^{3}=\left({x}^{2+5}\right)\cdot {x}^{3}={x}^{7}\cdot {x}^{3}={x}^{7+3}={x}^{10}[/latex], [latex]{x}^{2}\cdot {x}^{5}\cdot {x}^{3}={x}^{2+5+3}={x}^{10}[/latex], [latex]\begin{array}\text{ }\frac{y^{9}}{y^{5}}\hfill&=\frac{y\cdot y\cdot y\cdot y\cdot y\cdot y\cdot y}{y\cdot y\cdot y\cdot y\cdot y} \\ \hfill&=\frac{\cancel{y}\cdot\cancel{y}\cdot\cancel{y}\cdot\cancel{y}\cdot\cancel{y}\cdot y\cdot y\cdot y\cdot y}{\cancel{y}\cdot\cancel{y}\cdot\cancel{y}\cdot\cancel{y}\cdot\cancel{y}} \\ \hfill& =\frac{y\cdot y\cdot y\cdot y}{1} \\ \hfill& =y^{4}\end{array}[/latex], [latex]\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex], [latex]\frac{{y}^{9}}{{y}^{5}}={y}^{9 - 5}={y}^{4}[/latex]. Simplifying expressions mean rewriting the same algebraic expression with no like terms and in a compact manner. Check out all of our online calculators here! a1 n = na. Solve - Simplifying exponent expressions calculator Solve Simplify Factor Expand Graph GCF LCM Solve an equation, inequality or a system. a n = a a . When simplifying expressions with exponents, rather than trying to work robotically from the rules, instead think about what the exponents mean. Simplify Calculator. Here, there are two parentheses both having two unlike terms. Ok. that was just a quick review. It works with polynomials with more than one variable as well. 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And if there is a number or variable written just outside the bracket, then multiply it with all the terms inside using the distributive property. Since we have x^3 divided by x^7, we subtract their exponents. Simplify is the same as reducing to lowest terms when we talk about fractions. It is often simpler to work directly from the meaning of exponents. Simplify the math operation ie., on multiplying the two large exponents, we will get the final output. The simplified expression will only have unlike terms connected by addition/subtraction operators that cannot be simplified further. Choose "Simplify" from the topic selector and click to see the result in our Algebra Calculator! We need to learn how to simplify expressions as it allows us to work more efficiently with algebraic expressions and ease out our calculations. Completing a task step-by-step can help ensure that it is done correctly and efficiently. . Simplification can also help to improve your understanding of math concepts. But we know also ( 8 3) 3 = 8. We have shown that the exponential expression [latex]{a}^{n}[/latex] is defined when [latex]n[/latex] is a natural number, 0, or the negative of a natural number. flashcard sets. Expression Equation Inequality Contact us Simplify Factor Expand GCF LCM Our users: I purchased the Personal Algebra Tutor (PAT). If you need help, we're here for you 24/7. Do not simplify further. lessons in math, English, science, history, and more. Simplifying Expressions Calculator. Here are the basic steps to follow to simplify an algebraic expression: remove parentheses by multiplying factors use exponent rules to remove parentheses in terms with exponents combine like terms by adding coefficients combine the constants Let's work through an example. This video looks at how to work with expressions that have rational exponents (fractions in the exponent). Complex numbers involve the quantity known as i , an "imaginary" number with the property i = 1.If you have to simply an expression involving a complex number, it might seem daunting, but it's quite a simple process once you learn the basic rules. Well, 5 is positive, so we don't need to change it. Using b x b y = b x + y Simplify More ways to get app Simplify Calculator Since we have y ^8 divided by y ^3, we subtract their exponents. This calculator will solve your problems. In this case, we would use the zero exponent rule of exponents to simplify the expression to 1. Example 1: Find the simplified form of the expression formed by the following statement: "Addition of k and 8 multiplied by the subtraction of k from 16". . Please ensure that your password is at least 8 characters and contains each of the following: You'll be able to enter math problems once our session is over. There are rules in algebra for simplifying exponents with different and same bases that we can use. . Simplifying Radical Expressions replace the square root sign ( ) with the letter r. show help examples Preview: Input Expression: Examples: r125 8/r2 (1+2r2)^2 Step 2: Click "Simplify" to get a simplified version of the entered expression. Basic knowledge of algebraic expressions is required. By using the distributive property of simplifying expression, it can be simplified as. Otherwise, the difference [latex]m-n[/latex] could be zero or negative. Free Exponents Calculator - Simplify exponential expressions using algebraic rules step-by-step. Write each of the following products with a single base. Let's begin! This calculator will try to simplify a polynomial as much as possible. To simplify your expression using the Simplify Calculator, type in your expression like 2(5x+4)-3x. By using the product rule of exponents, it can be written as 2ab + 4b3 - 8ab, which is equal to 4b3 - 6ab. Example: Simplify the expression: 3/4x + y/2 (4x + 7). How to Simplify an Expression with Parentheses & Exponents, Power of Powers: Simplifying Exponential Expressions, Graphing Systems of Equations | Overview, Process & Examples, Negative Exponents: Writing Powers of Fractions and Decimals. For any real number [latex]a[/latex] and natural numbers [latex]m[/latex] and [latex]n[/latex], such that [latex]m>n[/latex], the quotient rule of exponents states that. For any nonzero real number [latex]a[/latex] and natural number [latex]n[/latex], the negative rule of exponents states that. Exponents Calculator Instructions for using FX Maths Pack. For any real number [latex]a[/latex] and positive integers [latex]m[/latex] and [latex]n[/latex], the power rule of exponents states that. copyright 2003-2023 Study.com. Free simplify calculator - simplify algebraic expressions step-by-step. (2) Adding, subtracting, multiplying and dividing of rational expressions (3) Simplify any expressions If you select option 1: Enter the numerator function Enter the denominator function Click 'calculate' If you choose option 2: First select either you need to apply operations on two or three expressions Therefore, we move the denominator to the numerator with a positive exponent : Now, we only have positive exponents and we can apply the product of exponents rule to simplify: To use the Simplify Calculator, simply enter your expression into the input field and press the "Calculate" button. Look at the above examples, and see whether and how we have used this property for the simplification of expressions. We know from our exponent properties that x^-4 is 1 / x^4 times y^5. With a negative exponent, this causes the expression to reciprocate and change exponent to positive, so start with 1/ (4096)^ (5/6) = 1/4^5 = 1/1024. The rules for exponential expressions can be combined to simplify more complicated expressions. Need help? Create your account, 13 chapters | Check out our online math support services! All rights reserved. Our final, simplified answer is y^5 / x^4. - Definition & Examples, Expressing Relationships as Algebraic Expressions, Practice Simplifying Algebraic Expressions, Expanding & Simplifying Algebraic Expressions, Translating an Addition Statement into an Algebraic Expression, Roots and Powers of Algebraic Expressions, Translating a Division Statement into an Algebraic Expression, Taking the Derivative of arcsin: How-To & Tutorial, Working Scholars Bringing Tuition-Free College to the Community. My last step is to multiply. Then the result is multiplied three times because the entire expression has an exponent of 3. This gives us y^8-3. Expressions can be rewritten using exponents to be simplified visually and mathematically. Let us take another example of simplifying 4(2a + 3a + 4) + 6b using the distributive property. There are many ways to improve your writing skills, but one of the most effective is to practice regularly. This gives us x^3-7. BYJU'S online simplifying The product [latex]8\cdot 16[/latex] equals 128, so the relationship is true. So, adding these two pairs of like terms will result in (6x - 3x) + (-x2 + x2). Check these interesting articles related to the concept of simplifying expressions in math. [latex]\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}[/latex]. Step 1, how do i find my safe credit union account number, how to write a number in expanded form in two ways, simplify expressions with rational exponents calculator. simplify rational or radical expressions with our free step-by-step math calculator. To simplify a power of a power, you multiply the exponents, keeping the base the same. In the denominator, I want the xs over each other and the ys over each other, so I write x^7y^3. Simplifying radical expressions calculator This calculator simplifies expressions that contain radicals. BYJU'S online simplifying. | 10 Free Exponents Calculator - Simplify exponential expressions using algebraic rules step-by-step. While simplifying expressions with fractions, we have to make sure that the fractions should be in the simplest form and only unlike terms should be present in the simplified expression. . [latex]\frac{t^{8}}{t^{8}}=\frac{\cancel{t^{8}}}{\cancel{t^{8}}}=1[/latex], If we were to simplify the original expression using the quotient rule, we would have. calculate equation by Improve your scholarly performance Definition 17.4.1: Rational Exponent a1 n. If na is a real number and n 2, then. This is the product rule of exponents. Step 2: Use the exponent rules to simplify terms containing exponents. See the steps to to. Do not simplify further. Step 3: Finally, the value of the given exponent will be displayed in the output field. Our community of experts can help you with any question you have. Here's an example: Enter 10, press the exponent key, then press 5 and enter. For example, 2x (x + y) can be simplified as 2x 2 + 2xy. Before you start making a list of calculations, however, you . For instance, a pixel is the smallest unit of light that can be perceived and recorded by a digital camera. When fractions are given in an expression, then we can use the distributive property and the exponent rules to simplify such expression. Along with PEMDAS, exponent rules, and the knowledge about operations on expressions also need to be used while simplifying algebraic expressions. Practice your math skills and learn step by step with our math solver. Our first step is to simplify (2p)^3. Step 1: Enter an exponential expression below which you want to simplify. Exponent Properties, Rules & Examples | What is an Exponent in Math? Factoring can help to make the expression more compact and easier to work with. Now, to multiply fractions, we multiply the numerators and the denominators separately. Let's keep simplifying. This same logic can be used for any positive integer exponent n to show that a 1 n = a n. RATIONAL EXPONENT a 1 n To unlock this lesson you must be a Study.com Member. Math problems can be determined by using a variety of methods. Open up brackets, if any. The simplification calculator allows you to take a simple or complex expression and simplify and reduce the expression to it's simplest form. When one piece is missing, it can be difficult to see the whole picture. On the other hand, x/2 + 1/2y is in a simplified form as fractions are in the reduced form and both are unlike terms. Algebra often involves simplifying expressions, but some expressions are more confusing to deal with than others. y^9 divided by y^9 is y^(9-9). How to Solve Exponents Download Article methods 1 Solving Basic Exponents 2 Adding, Subtracting and Multiplying Exponents 3 Solving Fractional Exponents Other Sections Related Articles References Article Summary Co-authored by David Jia Last Updated: February 27, 2023 Exponents are used when a number is multiplied by itself. For example, the expression 3x + 2y 4x + 5y can be simplified by combining like terms to get 3x 4x + 2y + 5y = -x + 7y. Type ^ for exponents like x^2 for x squared. That means that [latex]{a}^{n}[/latex] is defined for any integer [latex]n[/latex]. Therefore, - k2 + 8k + 128 is the simplified form of the given expression. This website uses cookies to ensure you get the best experience on our website. Some of the rules for simplifying expressions are listed below: To simplify expressions with exponents is done by applying the rules of exponents on the terms. And, Victoria bought 6 pencils each for $x, so the cost of 6 pencils = $6x. For example, lets look at the following example. Here is an example: 2x^2+x(4x+3) Need more problem types? It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule. Simplifying Expressions This section will provide several examples of how to simplify expressions with exponents including at least one problem about each property given above. If you want to improve your performance, you need to focus on your theoretical skills. 42 is 16. So, we will be solving the brackets first by multiplying x to the terms written inside. Both terms have the same base, x, but they are raised to different exponents. By simplifying it further, we will get 3x, which will be the final answer. The result is that [latex]{x}^{3}\cdot {x}^{4}={x}^{3+4}={x}^{7}[/latex]. In these cases, further simplification is not possible. Solution: From the given statement, the expression formed is (k + 8)(16 - k). Simplify expressions with negative exponents calculator - Apps can be a great way to help learners with their math. First, we open the brackets, if any. Exponent Calculator - Simplify Exponential Expression. For those who need an instant solution, we have the perfect answer. Are you tired of struggling with complex algebraic expressions? Mathematicians, scientists, and economists commonly encounter very large and very small numbers. The calculator displays 1.304596316E13. Exponent rules can be used to simplify terms with exponents. The exponent calculator simplifies the given exponential expression using the laws of exponents. Contains a great and useful calculator, this is one of the best apps relating to education no other app compares with this app it helped me to understand my work better it even shows how it was worked out I recommend to 7 of my friends and they are happy about this app. Find the total cost of buying pencils by both of them. My next step is to split these up using multiplication. System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval . How to Use the Negative If there is a negative sign just outside parentheses, change the sign of all the terms written inside that bracket to simplify it. Therefore, x (6 x) x (3 x) = 3x. We provide quick and easy solutions to all your homework problems. Expressions refer to mathematical statements having a minimum of two terms containing either numbers, variables, or both connected through an addition/subtraction operator in between. Do you find it hard to keep track of all the terms and constants in your equations? Click the blue arrow to submit. Try refreshing the page, or contact customer support. An error occurred trying to load this video. Explore the use of several properties used to simplify expressions with exponents, including the. Let's assume we are now not limited to whole numbers. Therefore, the total cost of pencils bought by them = $5x + $6x = $11x. Divide one exponential expression by another with a larger exponent. The goal of simplification is to make the expression easier to work with and understand, while still representing the same value. I can help you determine the answer to math problems. Free simplify calculator - simplify algebraic expressions step-by-step. Simplifying mathematical expressions implies rewriting the same algebraic statement compactly with no like terms. Now, combining all the terms will result in 6x - x2 - 3x + x2. EXAMPLE 1. Splitting the multiplication gives us x^3 / x^7 times y^8 / y^3. Give it a try now and see how it can simplify your algebraic expressions and make your math problems a breeze! If you're having problems memorizing these properties, I suggest using flash cards. ( ) You can have more time for your hobbies by making small changes to your daily routine. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. For an instance, (2/4)x + 3/6y is not the simplified expression, as fractions are not reduced to their lowest form. Overall, simplifying algebraic expressions is an important skill that can help you to save time, improve your understanding of math, and develop your problem-solving skills. Mathematics is a way of dealing with tasks that involves numbers and equations. The maximum possible number of bits of information used to film a one-hour (3,600-second) digital film is then an extremely large number. 16/8 is 2/1 times p^(1-3) times q^(2-4) times r^9. The calculator will show you all the steps and easy-to-understand explanations of how to simplify polynomials. Let's look at an, Count the number of triangles in the given figure, Describe all solutions in parametric vector form, How to find inverse trig functions without calculator, How to find the central angle of a sector calculator, How to find the short diagonal of a rhombus, Math examples of graphing x and y coordinate equations. Notice we get the same result by adding the three exponents in one step. Looking for support from expert professors? [latex]\begin{array}{ccc}\hfill \frac{{h}^{3}}{{h}^{5}}& =& \frac{h\cdot h\cdot h}{h\cdot h\cdot h\cdot h\cdot h}\hfill \\ & =& \frac{\cancel{h}\cdot \cancel{h}\cdot \cancel{h}}{\cancel{h}\cdot \cancel{h}\cdot \cancel{h}\cdot h\cdot h}\hfill \\ & =& \frac{1}{h\cdot h}\hfill \\ & =& \frac{1}{{h}^{2}}\hfill \end{array}[/latex], [latex]\begin{array}{ccc}\hfill \frac{{h}^{3}}{{h}^{5}}& =& {h}^{3 - 5}\hfill \\ & =& \text{ }{h}^{-2}\hfill \end{array}[/latex], [latex]\begin{array}{ccc}{a}^{-n}=\frac{1}{{a}^{n}}& \text{and}& {a}^{n}=\frac{1}{{a}^{-n}}\end{array}[/latex], [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex], [latex]\begin{array}{ccc}\hfill {\left(pq\right)}^{3}& =& \stackrel{3\text{ factors}}{{\left(pq\right)\cdot \left(pq\right)\cdot \left(pq\right)}}\hfill \\ & =& p\cdot q\cdot p\cdot q\cdot p\cdot q\hfill \\ & =& \stackrel{3\text{ factors}}{{p\cdot p\cdot p}}\cdot \stackrel{3\text{ factors}}{{q\cdot q\cdot q}}\hfill \\ & =& {p}^{3}\cdot {q}^{3}\hfill \end{array}[/latex], [latex]{\left(ab\right)}^{n}={a}^{n}{b}^{n}[/latex]. Simplifying Expressions Calculator is a free online tool that displays the simplification of the given algebraic expression. Get math help online by chatting with a tutor or watching a video lesson. 638+ Math Specialists 4.8/5 Quality score 85636+ Student Reviews Get Homework Help By following these steps, you should be able to simplify most algebraic expressions. Whether you are a student working on a math assignment or a professional dealing with equations as part of your job, the Simplify Expression Calculator is an essential tool that can save you time and make solving equations much easier. [latex]{\left({e}^{-2}{f}^{2}\right)}^{7}=\frac{{f}^{14}}{{e}^{14}}[/latex], [latex]\begin{array}{ccc}\hfill {\left({e}^{-2}{f}^{2}\right)}^{7}& =& {\left(\frac{{f}^{2}}{{e}^{2}}\right)}^{7}\hfill \\ & =& \frac{{f}^{14}}{{e}^{14}}\hfill \end{array}[/latex], [latex]\begin{array}{ccc}\hfill {\left({e}^{-2}{f}^{2}\right)}^{7}& =& {\left(\frac{{f}^{2}}{{e}^{2}}\right)}^{7}\hfill \\ & =& \frac{{\left({f}^{2}\right)}^{7}}{{\left({e}^{2}\right)}^{7}}\hfill \\ & =& \frac{{f}^{2\cdot 7}}{{e}^{2\cdot 7}}\hfill \\ & =& \frac{{f}^{14}}{{e}^{14}}\hfill \end{array}[/latex], [latex]{\left(\frac{a}{b}\right)}^{n}=\frac{{a}^{n}}{{b}^{n}}[/latex], CC licensed content, Specific attribution, http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2, http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1/Preface, [latex]\left(3a\right)^{7}\cdot\left(3a\right)^{10} [/latex], [latex]\left(\left(3a\right)^{7}\right)^{10} [/latex], [latex]\left(3a\right)^{7\cdot10} [/latex], [latex]{\left(a\cdot b\right)}^{n}={a}^{n}\cdot {b}^{n}[/latex], [latex]\left(-3\right)^{5}\cdot \left(-3\right)[/latex], [latex]{x}^{2}\cdot {x}^{5}\cdot {x}^{3}[/latex], [latex]{t}^{5}\cdot {t}^{3}={t}^{5+3}={t}^{8}[/latex], [latex]{\left(-3\right)}^{5}\cdot \left(-3\right)={\left(-3\right)}^{5}\cdot {\left(-3\right)}^{1}={\left(-3\right)}^{5+1}={\left(-3\right)}^{6}[/latex], [latex]{\left(\frac{2}{y}\right)}^{4}\cdot \left(\frac{2}{y}\right)[/latex], [latex]{t}^{3}\cdot {t}^{6}\cdot {t}^{5}[/latex], [latex]{\left(\frac{2}{y}\right)}^{5}[/latex], [latex]\frac{{\left(-2\right)}^{14}}{{\left(-2\right)}^{9}}[/latex], [latex]\frac{{\left(z\sqrt{2}\right)}^{5}}{z\sqrt{2}}[/latex], [latex]\frac{{\left(-2\right)}^{14}}{{\left(-2\right)}^{9}}={\left(-2\right)}^{14 - 9}={\left(-2\right)}^{5}[/latex], [latex]\frac{{t}^{23}}{{t}^{15}}={t}^{23 - 15}={t}^{8}[/latex], [latex]\frac{{\left(z\sqrt{2}\right)}^{5}}{z\sqrt{2}}={\left(z\sqrt{2}\right)}^{5 - 1}={\left(z\sqrt{2}\right)}^{4}[/latex], [latex]\frac{{\left(-3\right)}^{6}}{-3}[/latex], [latex]\frac{{\left(e{f}^{2}\right)}^{5}}{{\left(e{f}^{2}\right)}^{3}}[/latex], [latex]{\left(e{f}^{2}\right)}^{2}[/latex], [latex]{\left({x}^{2}\right)}^{7}[/latex], [latex]{\left({\left(2t\right)}^{5}\right)}^{3}[/latex], [latex]{\left({\left(-3\right)}^{5}\right)}^{11}[/latex], [latex]{\left({x}^{2}\right)}^{7}={x}^{2\cdot 7}={x}^{14}[/latex], [latex]{\left({\left(2t\right)}^{5}\right)}^{3}={\left(2t\right)}^{5\cdot 3}={\left(2t\right)}^{15}[/latex], [latex]{\left({\left(-3\right)}^{5}\right)}^{11}={\left(-3\right)}^{5\cdot 11}={\left(-3\right)}^{55}[/latex], [latex]{\left({\left(3y\right)}^{8}\right)}^{3}[/latex], [latex]{\left({t}^{5}\right)}^{7}[/latex], [latex]{\left({\left(-g\right)}^{4}\right)}^{4}[/latex], [latex]\frac{{\left({j}^{2}k\right)}^{4}}{\left({j}^{2}k\right)\cdot {\left({j}^{2}k\right)}^{3}}[/latex], [latex]\frac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}[/latex], [latex]\begin{array}\text{ }\frac{c^{3}}{c^{3}} \hfill& =c^{3-3} \\ \hfill& =c^{0} \\ \hfill& =1\end{array}[/latex], [latex]\begin{array}{ccc}\hfill \frac{-3{x}^{5}}{{x}^{5}}& =& -3\cdot \frac{{x}^{5}}{{x}^{5}}\hfill \\ & =& -3\cdot {x}^{5 - 5}\hfill \\ & =& -3\cdot {x}^{0}\hfill \\ & =& -3\cdot 1\hfill \\ & =& -3\hfill \end{array}[/latex], [latex]\begin{array}{cccc}\hfill \frac{{\left({j}^{2}k\right)}^{4}}{\left({j}^{2}k\right)\cdot {\left({j}^{2}k\right)}^{3}}& =& \frac{{\left({j}^{2}k\right)}^{4}}{{\left({j}^{2}k\right)}^{1+3}}\hfill & \text{Use the product rule in the denominator}.\hfill \\ & =& \frac{{\left({j}^{2}k\right)}^{4}}{{\left({j}^{2}k\right)}^{4}}\hfill & \text{Simplify}.\hfill \\ & =& {\left({j}^{2}k\right)}^{4 - 4}\hfill & \text{Use the quotient rule}.\hfill \\ & =& {\left({j}^{2}k\right)}^{0}\hfill & \text{Simplify}.\hfill \\ & =& 1& \end{array}[/latex], [latex]\begin{array}{cccc}\hfill \frac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}& =& 5{\left(r{s}^{2}\right)}^{2 - 2}\hfill & \text{Use the quotient rule}.\hfill \\ & =& 5{\left(r{s}^{2}\right)}^{0}\hfill & \text{Simplify}.\hfill \\ & =& 5\cdot 1\hfill & \text{Use the zero exponent rule}.\hfill \\ & =& 5\hfill & \text{Simplify}.\hfill \end{array}[/latex], [latex]\frac{{\left(d{e}^{2}\right)}^{11}}{2{\left(d{e}^{2}\right)}^{11}}[/latex], [latex]\frac{{w}^{4}\cdot {w}^{2}}{{w}^{6}}[/latex], [latex]\frac{{t}^{3}\cdot {t}^{4}}{{t}^{2}\cdot {t}^{5}}[/latex], [latex]\frac{{\theta }^{3}}{{\theta }^{10}}[/latex], [latex]\frac{{z}^{2}\cdot z}{{z}^{4}}[/latex], [latex]\frac{{\left(-5{t}^{3}\right)}^{4}}{{\left(-5{t}^{3}\right)}^{8}}[/latex], [latex]\frac{{\theta }^{3}}{{\theta }^{10}}={\theta }^{3 - 10}={\theta }^{-7}=\frac{1}{{\theta }^{7}}[/latex], [latex]\frac{{z}^{2}\cdot z}{{z}^{4}}=\frac{{z}^{2+1}}{{z}^{4}}=\frac{{z}^{3}}{{z}^{4}}={z}^{3 - 4}={z}^{-1}=\frac{1}{z}[/latex], [latex]\frac{{\left(-5{t}^{3}\right)}^{4}}{{\left(-5{t}^{3}\right)}^{8}}={\left(-5{t}^{3}\right)}^{4 - 8}={\left(-5{t}^{3}\right)}^{-4}=\frac{1}{{\left(-5{t}^{3}\right)}^{4}}[/latex], [latex]\frac{{\left(-3t\right)}^{2}}{{\left(-3t\right)}^{8}}[/latex], [latex]\frac{{f}^{47}}{{f}^{49}\cdot f}[/latex], [latex]\frac{1}{{\left(-3t\right)}^{6}}[/latex], [latex]{\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}[/latex], [latex]\frac{-7z}{{\left(-7z\right)}^{5}}[/latex], [latex]{b}^{2}\cdot {b}^{-8}={b}^{2 - 8}={b}^{-6}=\frac{1}{{b}^{6}}[/latex], [latex]{\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}={\left(-x\right)}^{5 - 5}={\left(-x\right)}^{0}=1[/latex], [latex]\frac{-7z}{{\left(-7z\right)}^{5}}=\frac{{\left(-7z\right)}^{1}}{{\left(-7z\right)}^{5}}={\left(-7z\right)}^{1 - 5}={\left(-7z\right)}^{-4}=\frac{1}{{\left(-7z\right)}^{4}}[/latex], [latex]\frac{{25}^{12}}{{25}^{13}}[/latex], [latex]{t}^{-5}=\frac{1}{{t}^{5}}[/latex], [latex]{\left(a{b}^{2}\right)}^{3}[/latex], [latex]{\left(-2{w}^{3}\right)}^{3}[/latex], [latex]\frac{1}{{\left(-7z\right)}^{4}}[/latex], [latex]{\left({e}^{-2}{f}^{2}\right)}^{7}[/latex], [latex]{\left(a{b}^{2}\right)}^{3}={\left(a\right)}^{3}\cdot {\left({b}^{2}\right)}^{3}={a}^{1\cdot 3}\cdot {b}^{2\cdot 3}={a}^{3}{b}^{6}[/latex], [latex]2{t}^{15}={\left(2\right)}^{15}\cdot {\left(t\right)}^{15}={2}^{15}{t}^{15}=32,768{t}^{15}[/latex], [latex]{\left(-2{w}^{3}\right)}^{3}={\left(-2\right)}^{3}\cdot {\left({w}^{3}\right)}^{3}=-8\cdot {w}^{3\cdot 3}=-8{w}^{9}[/latex], [latex]\frac{1}{{\left(-7z\right)}^{4}}=\frac{1}{{\left(-7\right)}^{4}\cdot {\left(z\right)}^{4}}=\frac{1}{2,401{z}^{4}}[/latex], [latex]{\left({e}^{-2}{f}^{2}\right)}^{7}={\left({e}^{-2}\right)}^{7}\cdot {\left({f}^{2}\right)}^{7}={e}^{-2\cdot 7}\cdot {f}^{2\cdot 7}={e}^{-14}{f}^{14}=\frac{{f}^{14}}{{e}^{14}}[/latex], [latex]{\left({g}^{2}{h}^{3}\right)}^{5}[/latex], [latex]{\left(-3{y}^{5}\right)}^{3}[/latex], [latex]\frac{1}{{\left({a}^{6}{b}^{7}\right)}^{3}}[/latex], [latex]{\left({r}^{3}{s}^{-2}\right)}^{4}[/latex], [latex]\frac{1}{{a}^{18}{b}^{21}}[/latex], [latex]{\left(\frac{4}{{z}^{11}}\right)}^{3}[/latex], [latex]{\left(\frac{p}{{q}^{3}}\right)}^{6}[/latex], [latex]{\left(\frac{-1}{{t}^{2}}\right)}^{27}[/latex], [latex]{\left({j}^{3}{k}^{-2}\right)}^{4}[/latex], [latex]{\left({m}^{-2}{n}^{-2}\right)}^{3}[/latex], [latex]{\left(\frac{4}{{z}^{11}}\right)}^{3}=\frac{{\left(4\right)}^{3}}{{\left({z}^{11}\right)}^{3}}=\frac{64}{{z}^{11\cdot 3}}=\frac{64}{{z}^{33}}[/latex], [latex]{\left(\frac{p}{{q}^{3}}\right)}^{6}=\frac{{\left(p\right)}^{6}}{{\left({q}^{3}\right)}^{6}}=\frac{{p}^{1\cdot 6}}{{q}^{3\cdot 6}}=\frac{{p}^{6}}{{q}^{18}}[/latex], [latex]{\\left(\frac{-1}{{t}^{2}}\\right)}^{27}=\frac{{\\left(-1\\right)}^{27}}{{\\left({t}^{2}\\right)}^{27}}=\frac{-1}{{t}^{2\cdot 27}}=\frac{-1}{{t}^{54}}=-\frac{1}{{t}^{54}}[/latex], [latex]{\left({j}^{3}{k}^{-2}\right)}^{4}={\left(\frac{{j}^{3}}{{k}^{2}}\right)}^{4}=\frac{{\left({j}^{3}\right)}^{4}}{{\left({k}^{2}\right)}^{4}}=\frac{{j}^{3\cdot 4}}{{k}^{2\cdot 4}}=\frac{{j}^{12}}{{k}^{8}}[/latex], [latex]{\left({m}^{-2}{n}^{-2}\right)}^{3}={\left(\frac{1}{{m}^{2}{n}^{2}}\right)}^{3}=\frac{{\left(1\right)}^{3}}{{\left({m}^{2}{n}^{2}\right)}^{3}}=\frac{1}{{\left({m}^{2}\right)}^{3}{\left({n}^{2}\right)}^{3}}=\frac{1}{{m}^{2\cdot 3}\cdot {n}^{2\cdot 3}}=\frac{1}{{m}^{6}{n}^{6}}[/latex], [latex]{\left(\frac{{b}^{5}}{c}\right)}^{3}[/latex], [latex]{\left(\frac{5}{{u}^{8}}\right)}^{4}[/latex], [latex]{\left(\frac{-1}{{w}^{3}}\right)}^{35}[/latex], [latex]{\left({p}^{-4}{q}^{3}\right)}^{8}[/latex], [latex]{\left({c}^{-5}{d}^{-3}\right)}^{4}[/latex], [latex]\frac{1}{{c}^{20}{d}^{12}}[/latex], [latex]{\left(6{m}^{2}{n}^{-1}\right)}^{3}[/latex], [latex]{17}^{5}\cdot {17}^{-4}\cdot {17}^{-3}[/latex], [latex]{\left(\frac{{u}^{-1}v}{{v}^{-1}}\right)}^{2}[/latex], [latex]\left(-2{a}^{3}{b}^{-1}\right)\left(5{a}^{-2}{b}^{2}\right)[/latex], [latex]{\left({x}^{2}\sqrt{2}\right)}^{4}{\left({x}^{2}\sqrt{2}\right)}^{-4}[/latex], [latex]\frac{{\left(3{w}^{2}\right)}^{5}}{{\left(6{w}^{-2}\right)}^{2}}[/latex], [latex]\begin{array}{cccc}\hfill {\left(6{m}^{2}{n}^{-1}\right)}^{3}& =& {\left(6\right)}^{3}{\left({m}^{2}\right)}^{3}{\left({n}^{-1}\right)}^{3}\hfill & \text{The power of a product rule}\hfill \\ & =& {6}^{3}{m}^{2\cdot 3}{n}^{-1\cdot 3}\hfill & \text{The power rule}\hfill \\ & =& \text{ }216{m}^{6}{n}^{-3}\hfill & \text{Simplify}.\hfill \\ & =& \frac{216{m}^{6}}{{n}^{3}}\hfill & \text{The negative exponent rule}\hfill \end{array}[/latex], [latex]\begin{array}{cccc}\hfill {17}^{5}\cdot {17}^{-4}\cdot {17}^{-3}& =& {17}^{5 - 4-3}\hfill & \text{The product rule}\hfill \\ & =& {17}^{-2}\hfill & \text{Simplify}.\hfill \\ & =& \frac{1}{{17}^{2}}\text{ or }\frac{1}{289}\hfill & \text{The negative exponent rule}\hfill \end{array}[/latex], [latex]\begin{array}{cccc}\hfill {\left(\frac{{u}^{-1}v}{{v}^{-1}}\right)}^{2}& =& \frac{{\left({u}^{-1}v\right)}^{2}}{{\left({v}^{-1}\right)}^{2}}\hfill & \text{The power of a quotient rule}\hfill \\ & =& \frac{{u}^{-2}{v}^{2}}{{v}^{-2}}\hfill & \text{The power of a product rule}\hfill \\ & =& {u}^{-2}{v}^{2-\left(-2\right)}& \text{The quotient rule}\hfill \\ & =& {u}^{-2}{v}^{4}\hfill & \text{Simplify}.\hfill \\ & =& \frac{{v}^{4}}{{u}^{2}}\hfill & \text{The negative exponent rule}\hfill \end{array}[/latex], [latex]\begin{array}{cccc}\hfill \left(-2{a}^{3}{b}^{-1}\right)\left(5{a}^{-2}{b}^{2}\right)& =& -2\cdot 5\cdot {a}^{3}\cdot {a}^{-2}\cdot {b}^{-1}\cdot {b}^{2}\hfill & \text{Commutative and associative laws of multiplication}\hfill \\ & =& -10\cdot {a}^{3 - 2}\cdot {b}^{-1+2}\hfill & \text{The product rule}\hfill \\ & =& -10ab\hfill & \text{Simplify}.\hfill \end{array}[/latex], [latex]\begin{array}{cccc}\hfill {\left({x}^{2}\sqrt{2}\right)}^{4}{\left({x}^{2}\sqrt{2}\right)}^{-4}& =& {\left({x}^{2}\sqrt{2}\right)}^{4 - 4}\hfill & \text{The product rule}\hfill \\ & =& \text{ }{\left({x}^{2}\sqrt{2}\right)}^{0}\hfill & \text{Simplify}.\hfill \\ & =& 1\hfill & \text{The zero exponent rule}\hfill \end{array}[/latex], [latex]\begin{array}{cccc}\hfill \frac{{\left(3{w}^{2}\right)}^{5}}{{\left(6{w}^{-2}\right)}^{2}}& =& \frac{{\left(3\right)}^{5}\cdot {\left({w}^{2}\right)}^{5}}{{\left(6\right)}^{2}\cdot {\left({w}^{-2}\right)}^{2}}\hfill & \text{The power of a product rule}\hfill \\ & =& \frac{{3}^{5}{w}^{2\cdot 5}}{{6}^{2}{w}^{-2\cdot 2}}\hfill & \text{The power rule}\hfill \\ & =& \frac{243{w}^{10}}{36{w}^{-4}}\hfill & \text{Simplify}.\hfill \\ & =& \frac{27{w}^{10-\left(-4\right)}}{4}\hfill & \text{The quotient rule and reduce fraction}\hfill \\ & =& \frac{27{w}^{14}}{4}\hfill & \text{Simplify}.\hfill \end{array}[/latex], [latex]{\left(2u{v}^{-2}\right)}^{-3}[/latex], [latex]{x}^{8}\cdot {x}^{-12}\cdot x[/latex], [latex]{\left(\frac{{e}^{2}{f}^{-3}}{{f}^{-1}}\right)}^{2}[/latex], [latex]\left(9{r}^{-5}{s}^{3}\right)\left(3{r}^{6}{s}^{-4}\right)[/latex], [latex]{\left(\frac{4}{9}t{w}^{-2}\right)}^{-3}{\left(\frac{4}{9}t{w}^{-2}\right)}^{3}[/latex], [latex]\frac{{\left(2{h}^{2}k\right)}^{4}}{{\left(7{h}^{-1}{k}^{2}\right)}^{2}}[/latex]. northumberland news obituaries,