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The simplify calculator will then show you the steps to, The power rule applies to exponents. Use the quotient rule to simplify each expression. Get detailed solutions to your math problems with our Combining like terms step-by-step calculator. How to Simplify Expressions with Exponents - Study.com Simplify the expression using the properties of exponents calculator - Solve equations with PEMDAS order of operations showing the work. Quality is important in all aspects of life. Simplify expressions with positive exponents calculator Factoring can help to make the expression more compact and easier to work with. If we equate the two answers, the result is [latex]{t}^{0}=1[/latex]. How to simplify the expression with exponents - Math Index There are a lot of letters and numbers here, but don't let them trick you. Question ID 14047, 14058, 14059, 14046, 14051, 14056, 14057.. We begin by using the associative and commutative properties of multiplication to regroup the factors. Simplify Calculator Exponents are supported on variables using the ^ (caret) symbol. Simplifying polynomials calculator - softmath Look at the image given below showing another simplifying expression example. Negative Exponents in Fractions - Rule and Examples - Mechamath Its like a teacher waved a magic wand and did the work for me. Simplify exponents expressions calculator | Math Workbook Simplify Calculator Simplifying expressions using rules of exponents calculator Simplify It can be very useful while simplifying expressions. Completing a task step-by-step can help ensure that it is done correctly and efficiently. Exponents Calculator Instructions for using FX Maths Pack. Step 2: Now click the button "Solve" to get the result. The best oart of it is, that it shows the steps to the solutions, not like a regular calculator, i suck at math and I'm a senior in high school about to go in junior year and this helps a lot considering teachers at my school aren't much help anyways, I'm just glad this app exists. How to simplify expressions with exponents calculator x(6 - x) can be simplified as 6x - x2, and -x(3 - x) can be simplified as -3x + x2. Simplify x.x2 Then we simplify the terms containing exponents. This will give us x^3-7, which is -4 and y^8-3, which is 5. The power of a product of factors is the same as the product of the powers of the same factors. Using a calculator, we enter [latex]2,048\times 1,536\times 48\times 24\times 3,600[/latex] and press ENTER. a1 n = na. Exponentiation is a mathematical operation, written as an, involving the base a and an exponent n. In the case where n is a positive integer, exponentiation corresponds to repeated multiplication of the base, n times. Typing Exponents Type ^ for exponents like x^2 for "x squared". For the time being, we must be aware of the condition [latex]m>n[/latex]. Use the properties of logarithms: If an expression contains logarithms, you can use the properties of logarithms to simplify it. 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. Check these interesting articles related to the concept of simplifying expressions in math. Simplify the expression using the properties of rational exponents algebra simplify division equations 6th grade Math TEKS chart source code of rational expression calculator algebraic rational expressions simplifying. Simplify each expression and write the answer with positive exponents only. In addition to its practical benefits, simplifying expressions is also a great way to develop your problem-solving skills. Whether you are a student working on math assignments or a professional dealing with equations as part of your job, learning to simplify expressions is a valuable investment in your mathematical education and career. Now, combining all the terms will result in 6x - x2 - 3x + x2. Work on the task that is enjoyable to you Mathematics is the study of numbers, shapes, and patterns. Looking for support from expert professors? Simplifying radical expressions calculator - mathportal.org Our expert tutors are available 24/7 to give you the answer you need in real-time. You can also use the calculator to check your work and ensure that you have correctly simplified your expression. Use the power rule to simplify each expression. We provide quick and easy solutions to all your homework problems. This calculator will solve your problems. Now, let us learn how to use the distributive property to simplify expressions with fractions. To do this, we use the power rule of exponents. Therefore, 4(2a + 3a + 4) + 6b is simplified as 20a + 6b + 16. What are the steps for simplifying expressions. Introduction Exponents can be attached to variables as well as numbers. simplify rational or radical expressions with our free step-by-step math calculator. Step 1: Enter the expression you want to simplify into the editor. 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. Simplifying radical expressions calculator This calculator simplifies expressions that contain radicals. Otherwise, the difference [latex]m-n[/latex] could be zero or negative. It requires one to be familiar with the concepts of arithmetic operations on algebraic expressions, fractions, and exponents. Step 2: Use the exponent rules to simplify terms containing exponents. Are you tired of struggling with complex algebraic expressions? Simplifying radical expressions (addition) | Algebra (video) - Khan Academy Get math help online by chatting with a tutor or watching a video lesson. This is amazing, it helped me so often already! Suppose an exponential expression is raised to some power. One of the main benefits of simplifying expressions is that it can save you time and effort. Expand each expression, and then rewrite the resulting expression. 16/8 is 2/1 times p^(1-3) times q^(2-4) times r^9. Solve - Properties of rational exponents calculator. Simplifying expressions with exponents calculator | Zens 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. If you're looking for a tutor who can help you with any subject, look no further than Instant Expert Tutoring. By using the distributive property, the given expression can be written as 3/4x + y/2 (4x) + y/2 (7). 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. In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents. Let's try the best Simplify expressions . Check out all of our online calculators here! I can help you with any mathematic task you need help with. Therefore, x (6 x) x (3 x) = 3x. Remove unnecessary terms: If a term has a coefficient of 0, it can be removed from the expression since it has no effect on the value. Simplify In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. If there is a negative sign outside the bracket, then remove the bracket and change the signs of all the terms written inside from + to -, and - to +. Next step - look at each part individually. . Let us take another example of simplifying 4(2a + 3a + 4) + 6b using the distributive property. In this equation, you'd start by simplifying the part of the expression in parentheses: 24 - 20. Simplify expressions with positive exponents calculator As a college student who struggles with algebra like, bUT SOMETIMES THERE ARE SOME PROBLEMS. 5.3: Simplify Rational Exponents - Mathematics LibreTexts As a member, you'll also get unlimited access to over 88,000 Some useful properties include. Since we have x^3 divided by x^7, we subtract their exponents. This tool is designed to take the frustration out of algebra by helping you to simplify and reduce your expressions to their simplest form. 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. The first step I like to do is put the like terms on top of each other. The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but different exponents. Math is the study of numbers, shapes, and patterns. It uses advanced algorithms to quickly and accurately combine like terms and eliminate unnecessary factors, giving you a clean, simplified expression that is much easier to work with. The calculator will simplify the equation step-by-step, and display the result. Negative Exponents Calculator - Free online Calculator - BYJUS Confidentiality is important in order to maintain trust between parties. Simplifies expressions step-by-step and shows the work! This website helped me pass! Simplify expressions using properties of exponents calculator Plus, get practice tests, quizzes, and personalized coaching to help you Step 2: Click "Simplify" to get a simplified version of the entered expression. EXAMPLE 1. If there is a positive sign outside the bracket, then remove the bracket and write all the terms retaining their original signs. Give it a try now and see how it can simplify your algebraic expressions and make your math problems a breeze! In this section, we review rules of exponents first and then apply them to calculations involving very large or small numbers. Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses, Factoring with FOIL, Graphing Parabolas and Solving Quadratics. Simplifying Expressions Calculator Exponents are supported on variables using the ^ (caret) symbol. 2 (24 - 20)2 + 18 / 6 - 30. We made the condition that [latex]m>n[/latex] so that the difference [latex]m-n[/latex] would never be zero or negative. 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. Expressions can be rewritten using exponents to be simplified visually and mathematically. Evaluating fractional exponents | Algebra (video) | Khan Academy Write answers with positive exponents. You can use the keyboard to enter exponents, fractions, and parentheses, among others. By using the distributive property of simplifying expression, it can be simplified as. Simplify exponential expressions calculator Try the Free Math Solver or Scroll down to Tutorials! Expand and simplify polynomials. I highly recommend you use this site! I feel like its a lifeline. Math understanding that gets you You can use math to determine all sorts of things, like how much money you'll need to save for a rainy day. In other words, [latex]{\left(pq\right)}^{3}={p}^{3}\cdot {q}^{3}[/latex]. First, we open the brackets, if any. Suppose we want to find a number p such that (8p)3 = 8. Rational Expressions Calculator - Simplify Numbers & Function It does not intend to find the value of an unknown quantity. Simplifying Expressions with Exponents - NROC In this blog post, we will be discussing How to simplify expressions with exponents calculator. This appears later in more advanced courses, but for now, we will consider the value to be undefined. Simplify Calculator. Exponent Calculator In just five seconds, you can get the answer to any question you have. Lets rewrite the original problem differently and look at the result. Free Exponents Calculator - Simplify exponential expressions using algebraic rules step-by-step. Simplifying expressions with exponents calculator free a. n times. At first, it may appear that we cannot simplify a product of three factors. In math, simplifying expressions is a way to write an expression in its lowest form by combining all like terms together. Simplifying Expressions Calculator is a free online tool that displays the simplification of the given algebraic expression. 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We are asked to simplify using positive exponents: p^(-2) is the same as 1/p^2; q^(-2) is the same 1/q^2. ti 89 algebra discovery distributive property nc discrete math practice problems rational expressions calculator using excel to find least common number from Easily simplify expressions containing exponents | Purplemath . Simplify each expression using the zero exponent rule of exponents. Simplify (x-2x-3)4. Sort by: Top Voted Questions Tips & Thanks Exponents are supported on variables using the ^ (caret) symbol. Typing Exponents. 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. Simplify Calculator - MathPapa When we use rational exponents, we can apply the properties of exponents to simplify expressions. Kathryn teaches college math. This gives us x^3-7. Type ^ for exponents like x^2 for "x squared". [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].