Step 3: Finally, the value of the given exponent will be displayed in the output field. What Are the Five Main Exponent Properties? Type ^ for exponents like x^2 for x squared. calculate equation by Improve your scholarly performance It works with polynomials with more than one variable as well. Mathematicians, scientists, and economists commonly encounter very large and very small numbers. Simplifying Expressions with Distributive Property, Addition and subtraction of algebraic expressions. Since we have y^8 divided by y^3, we subtract their exponents. The rules for exponents may be combined to simplify expressions. This calculator will allow compute an simplify numeric expressions that involve exponents. Expressions can be rewritten using exponents to be simplified visually and mathematically. Example: 2x-1=y,2y+3=x New Example Keyboard Solve e i s c t l L Search Engine users found our website today by entering these keyword phrases : Write answers with positive exponents. Free Exponents Calculator - Simplify exponential expressions using algebraic rules step-by-step. Now consider an example with real numbers. Simplification can also help to improve your understanding of math concepts. Suppose we want to find a number p such that (8p)3 = 8. Math understanding that gets you This is, you work on parentheses first, then on the exponents, then you do the multiplications and so on. Step 2: Now click the button "Solve" to get the result. There are a lot of letters and numbers here, but don't let them trick you. Look at the above examples, and see whether and how we have used this property for the simplification of expressions. 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. Get math help online by chatting with a tutor or watching a video lesson. Math problems can be determined by using a variety of methods. What are the steps for simplifying expressions. Math is the study of numbers, shapes, and patterns. Exponents The calculator allows with this computer algebra function of reducing an algebraic expression. In this section, we review rules of exponents first and then apply them to calculations involving very large or small numbers. We follow the same PEMDAS rule to simplify algebraic expressions as we do for simple arithmetic expressions. . Sort by: Top Voted Questions Tips & Thanks [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. Multi-Step Equations with Fractions & Decimals | Solving Equations with Fractions. So, adding these two pairs of like terms will result in (6x - 3x) + (-x2 + x2). Use the zero exponent and other rules to simplify each expression. For an instance, (2/4)x + 3/6y is not the simplified expression, as fractions are not reduced to their lowest form. . Simplify each expression and write the answer with positive exponents only. The cost of all 5 pencils = $5x. The equations section lets you solve an equation or system of equations. This is the product rule of exponents. Here is an example: 2x^2+x(4x+3) Need more problem types? By learning to identify patterns and relationships, and by using the properties of exponents and logarithms to simplify expressions, you can improve your ability to think critically and solve complex problems. . Simplify (m14n12)2(m2n3)12 To simplify algebraic expressions, follow the steps given below: Step 1: Solve parentheses by adding/subtracting like terms inside and by multiplying the terms inside the brackets with the factor written outside. 2 42 + 18 / 6 - 30. The Power Property for Exponents says that (am)n = am n when m and n are whole numbers. 986+ Experts. To simplify your expression using the Simplify Calculator, type in your expression like 2 (5x+4)-3x. Simplify mathematical expressions involving addition, subtraction, multiplication, division, and exponents Simplify Expressions Using the Order of Operations We've introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. For example, you can combine 3x and 2x by adding them to get 3x + 2x = 5x. Simplify the expression: x (6 x) x (3 x). The "Exponents" calculator is great for those with a basic understanding of exponents. When one piece is missing, it can be difficult to see the whole picture. Simplify Notice that the exponent of the product is the sum of the exponents of the terms. Get unlimited access to over 88,000 lessons. When you enter an expression into the calculator, the calculator will simplify the Exponents are supported on variables using the ^ (caret) symbol. 2 2 = 2 2 = 4 Square Root Calculator Calculate real and complex square roots (2nd order roots) of numbers or x. Follow the PEMDAS rule to determine the order of terms to be simplified in an expression. What does this mean? If you wish to solve the equation, use the Equation Solving Calculator. To use the Simplify Calculator, simply enter your expression into the input field and press the Calculate button. In this article, we will be focussing more on how to simplify algebraic expressions. Therefore, 4(2a + 3a + 4) + 6b is simplified as 20a + 6b + 16. For example, can we simplify [latex]\frac{{h}^{3}}{{h}^{5}}[/latex]? Multiplying straight across, our final answer is 1/3x^2. Now let's look at a couple of examples! This time we have 5x^2y^9 / 15y^9x^4. 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. 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. Along with PEMDAS, exponent rules, and the knowledge about operations on expressions also need to be used while simplifying algebraic expressions. Therefore, - k2 + 8k + 128 is the simplified form of the given expression. It helped me pass my exam and the test questions are very similar to the practice quizzes on Study.com. We begin by using the associative and commutative properties of multiplication to regroup the factors. This calculator will try to simplify a polynomial as much as possible. Multiply the exponents on the left.Write the exponent 1 on the right.Since the bases are the same, the exponents must be equal.Solve for p. So ( 8 1 3) 3 = 8. . Another useful result occurs if we relax the condition that [latex]m>n[/latex] in the quotient rule even further. We can always check that this is true by simplifying each exponential expression. The simplification calculator allows you to take a simple or complex expression and simplify and reduce the expression to it's simplest form. Our support team is available 24/7 to assist you. Various arithmetic operations like addition, subtraction, multiplication, and division can be applied to simplify . Simplifying these terms using positive exponents makes it even easier for us to read. Look at the image given below showing another simplifying expression example. It can be very useful while simplifying expressions. Typing Exponents. What would happen if [latex]m=n[/latex]? Before you start making a list of calculations, however, you . In just five seconds, you can get the answer to any question you have. Solve Now How to Simplify Exponents or Powers on the TI 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. BYJU'S online negative exponents calculator tool makes the calculation faster, and it displays the result in a fraction of seconds. If we keep separating the terms and following the properties, we'll be fine. You need to provide a valid expression that involves exponents. Factor the expression: Factoring an expression involves identifying common factors among the terms and pulling them out of the expression using parentheses. All three are unlike terms, so it is the simplified form of the given expression. Explore the use of several properties used to simplify expressions with exponents, including the. . How to Use the Negative 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. For instance, a pixel is the smallest unit of light that can be perceived and recorded by a digital camera. Solution: Given, Daniel bought 5 pencils each for $x. As a member, you'll also get unlimited access to over 88,000 Simplify 2n(n2+3n+4) To see how this is done, let us begin with an example. To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. Simplifying algebraic expressions is a fundamental skill that is essential for anyone working with math, whether you are a student or a professional. Simplifying expressions mean rewriting the same algebraic expression with no like terms and in a compact manner. When fractions are given in an expression, then we can use the distributive property and the exponent rules to simplify such expression. The result is that [latex]{x}^{3}\cdot {x}^{4}={x}^{3+4}={x}^{7}[/latex]. Let's begin! Created by Sal Khan and Monterey Institute for Technology and Education. Indulging in rote learning, you are likely to forget concepts. There will be times when working with expressions will be easier if you use rational exponents and times when it will be easier if you use radicals. Simplify expressions with positive exponents calculator - Math can be a challenging subject for many learners. Expressions can be rewritten using exponents to be simplified visually and mathematically. Do not simplify further. Consider the expression [latex]{\left({x}^{2}\right)}^{3}[/latex]. To simplify a power of a power, you multiply the exponents, keeping the base the same. If so, then you will love the Simplify Calculator. Simplify Calculator Exponents are supported on variables using the ^ (caret) symbol. Solutions Graphing Practice; New Geometry; Calculators; Notebook . This calculator will solve your problems. Enrolling in a course lets you earn progress by passing quizzes and exams. This is true for any nonzero real number, or any variable representing a real number. Here is an example: 2x^2+x (4x+3) Confidentiality is important in order to maintain trust between parties. By using the distributive property of simplifying expression, it can be simplified as. But there is support available in the form of. For example, lets look at the following example. You can improve your academic performance by studying regularly and attending class. In this example, we simplify (2x)+48+3 (2x)+8. By simplifying the expression, you can eliminate unnecessary terms and constants, making it easier to focus on the important parts of the equation and work through it step by step. Yes. The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but different exponents. Expand and simplify polynomials. For any real numbers [latex]a[/latex] and [latex]b[/latex], where [latex]b\neq0[/latex], and any integer [latex]n[/latex], the power of a 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. It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule. Step 2: Click "Simplify" to get a simplified version of the entered expression. This simplified expression is equivalent to the original one, but it is written in a simpler and more compact form. If you need more of a review, please go back and review the entire lesson! See the steps to to. There are many ways to improve your writing skills, but one of the most effective is to practice regularly. Do you find it hard to keep track of all the terms and constants in your equations? As a college student who struggles with algebra like, bUT SOMETIMES THERE ARE SOME PROBLEMS. Simplify radical,rational expression with Step. Write answers with positive exponents. You can improve your educational performance by studying regularly and practicing good study habits. While the "Fractional Exponents" calculator and "Solve for Exponents" calculator, assist those with a more advanced understanding of exponents. Ok. that was just a quick review. I would definitely recommend Study.com to my colleagues. Therefore, x (6 x) x (3 x) = 3x. Check out our online math support services! Perform the division by canceling common factors. 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 Simplifying algebraic expressions refer to the process of reducing the expression to its lowest form. Simplify Calculator Simplify algebraic expressions step-by-step full pad Examples Related Symbolab blog posts Just like numbers have factors (23=6), expressions have factors ( ` . Looking for support from expert professors? Check out. Notice we get the same result by adding the three exponents in one step. [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].
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