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This typically involves combining like terms (terms with the same variables and exponents), removing unnecessary constants or terms, and rearranging the expression in a more convenient form. Practice your math skills and learn step by step with our math solver. Doing math equations is a great way to keep your mind sharp and improve your problem-solving skills. Solution: By using the rules of simplifying expressions, 4ps - 2s - 3(ps +1) - 2s can be simplified as. Free simplify calculator - simplify algebraic expressions step-by-step. Simplify Calculator. You can improve your educational performance by studying regularly and practicing good study habits. Write answers with positive exponents. We begin by using the associative and commutative properties of multiplication to regroup the factors. Completing a task step-by-step can help ensure that it is done correctly and efficiently. Simplify the math operation ie., on multiplying the two large exponents, we will get the final output. System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval . This section will provide several examples of how to simplify expressions with exponents including at least one problem about each property given above. ti 89 algebra discovery distributive property nc discrete math practice problems rational expressions calculator using excel to find least common number from Simplifying expressions with exponents In the term , is the base and is the exponent. 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. Explore the use of several properties used to simplify expressions with exponents, including the. I can help you determine the answer to math problems. . 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. The simplification calculator allows you to take a simple or complex expression and simplify and reduce the expression to it's simplest form. Simplifying Expressions Calculator Exponents are supported on variables using the ^ (caret) symbol. On most calculators, you enter the base, press the exponent key and enter the exponent. Let us take another example of simplifying 4(2a + 3a + 4) + 6b using the distributive property. However, when simplifying expressions containing exponents, don't feel like you must work only with, or straight from, these rules. For any real number [latex]a[/latex] and positive integers [latex]m[/latex] and [latex]n[/latex], the power rule of exponents states that. . . Here, there are two parentheses both having two unlike terms. Groups Cheat . We follow the same PEMDAS rule to simplify algebraic expressions as we do for simple arithmetic expressions. This is, you work on parentheses first, then on the exponents, then you do the multiplications and so on. Use the quotient rule to simplify each expression. For instance, a pixel is the smallest unit of light that can be perceived and recorded by a digital camera. 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. BYJU'S online negative exponents calculator tool makes the calculation faster, and it displays the result in a fraction of seconds. Mathematics is a way of dealing with tasks that involves numbers and equations. The basic rule for simplifying expressions is to combine like terms together and write unlike terms as it is. . Simplifying Exponents. We know from our exponent properties that x^-4 is 1 / x^4 times y^5. copyright 2003-2023 Study.com. Distributive property can be used to simplify the. In the term , is the base and is the exponent. We made the condition that [latex]m>n[/latex] so that the difference [latex]m-n[/latex] would never be zero or negative. Simplifying radical expressions calculator Free radical equation calculator - solve radical equations step-by-step. Simplifying mathematical expressions implies rewriting the same algebraic statement compactly with no like terms. If we keep separating the terms and following the properties, we'll be fine. When using the product rule, different terms with the same bases are raised to exponents. Simplify x.x2 To simplify an algebraic expression means to rewrite it in a simpler form, without changing its value. The calculator above accepts negative bases, but does not compute imaginary numbers. But there is support available in the form of. Then it must be that ( 8 1 3) 3 = 8 3. Simplify Expressions With Negative Exponents. When using the power rule, a term in exponential notation is raised to a power. Simplify the expression using the properties of exponents calculator - Solve equations with PEMDAS order of operations showing the work. 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. Exponents Multiplying straight across, our final answer is 1/3x^2. I can help you with any mathematic task you need help with. The general rule to simplify expressions is PEMDAS - stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. In this expression, 6x and -3x are like terms, and -x2 and x2 are like terms. [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. It works with polynomials with more than one variable as well. In this article, we will be focussing more on how to simplify algebraic expressions. When simplifying expressions with exponents, rather than trying to work robotically from the rules, instead think about what the exponents mean. Notice that the exponent of the quotient is the difference between the exponents of the divisor and dividend. Looking for a quick and easy way to get help with your homework? Used with the function expand, the function simplify can expand and collapse a literal expression. As a college student who struggles with algebra like, bUT SOMETIMES THERE ARE SOME PROBLEMS. For example, to express x2, enter x^2. 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. This gives us x^3-7. See the steps to to. algebra simplify division equations 6th grade Math TEKS chart source code of rational expression calculator algebraic rational expressions simplifying. Looking for help with your math homework? For any nonzero real number [latex]a[/latex] and natural number [latex]n[/latex], the negative rule of exponents states that. Simplifying expressions mean rewriting the same algebraic expression with no like terms and in a compact manner. Also, the product and quotient rules and all of the rules we will look at soon hold for any integer [latex]n[/latex]. Ok. that was just a quick review. Let's try the best Simplify expressions . Use the distributive property to multiply any two polynomials. Mathway requires javascript and a modern browser. The simplify calculator will then show you the steps to help you learn how to simplify your algebraic expression on your own. The denominator of the rational exponent is the index of the radical. . This is amazing, it helped me so often already! When you are working with a simplified expression, it is easier to see the underlying patterns and relationships that govern the equation. To simplify your expression using the Simplify Calculator, type in your expression like 2(5x+4)-3x. When one piece is missing, it can be difficult to see the whole picture. Step 1: Enter an exponential expression below which you want to simplify. Write each of the following products with a single base. Confidentiality is important in order to maintain trust between parties. Need help? Our first expression has x^3y^8 / y^3x^7. This tool is designed to take the frustration out of algebra by helping you to simplify and reduce your expressions to their simplest form. In these cases, further simplification is not possible. For example, can we simplify [latex]\frac{{h}^{3}}{{h}^{5}}[/latex]? In a similar way to the product rule, we can simplify an expression such as [latex]\frac{{y}^{m}}{{y}^{n}}[/latex], where [latex]m>n[/latex]. Now, to multiply fractions, we multiply the numerators and the denominators separately. In this case, you add the exponents. - 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. lessons in math, English, science, history, and more. Now let's look at a couple of examples! Simplify the expression \frac { { { {x}^ {2}}}} { { { {x}^ { {-3}}}}} x3x2. This is in simplified form using positive exponents. BYJU'S online simplifying. 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. Simplify an expression or cancel an expression means reduce it by grouping terms. 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. If you need help, we're here for you 24/7. 5/15 reduces to 1/3. Before learning about simplifying expressions, let us quickly go through the meaning of expressions in math. There's one exponent in this equation: 42, or four to the second power. simplify rational or radical expressions with our free step-by-step math calculator. The "Exponents" calculator is great for those with a basic understanding of exponents. An expression with a negative exponent is defined as a reciprocal. 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. Step 1: Enter the algebraic expression in the corresponding input box. 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. Simplify, Simplify (a12b)12(ab12) Simplifying dividing algebraic expressions, solve 3x3 systems of linear equations with TI-84 calculator, solving parabola functions, Easiest way to Factor a third-degree polynomial. What does this mean? So, adding these two pairs of like terms will result in (6x - 3x) + (-x2 + x2). simplify rational or radical expressions with our free step-by-step math calculator. Simplify each expression and write the answer with positive exponents only. 2 2 = 2 2 = 4 Square Root Calculator Calculate real and complex square roots (2nd order roots) of numbers or x. . Putting the answers together, we have [latex]{h}^{-2}=\frac{1}{{h}^{2}}[/latex]. Both terms have the same base, x, but they are raised to different exponents. Determine mathematic problems Determining mathematical problems can be difficult, but with practice it can become easier. This calculator will solve your problems. To find the product of powersMultiplication of two or more values in exponential form that have the same base-. Factoring can help to make the expression more compact and easier to work with. In this blog post, we will be discussing How to simplify expressions with exponents calculator. Flash cards are a fantastic and easy way to memorize topics, especially math properties. 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. Next step - look at each part individually. 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. Are you tired of struggling with complex algebraic expressions? Notice we get the same result by adding the three exponents in one step. This calculator will allow compute an simplify numeric expressions that involve exponents. There are many ways to improve your writing skills, but one of the most effective is to practice regularly. Next, we separate them into multiplication: 16/8 times p/p^3 times q^2 / q^4 times r^9. 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. Write each of the following products with a single base. Solve an equation, inequality or a system. Volume & Surface Area of a Sphere | How to Find the Surface Area of a Sphere, System of Equations Word Problems & Explanations | How to Solve System of Equations Word Problems, Negative Signs and Simplifying Algebraic Expressions, SAT Subject Test Mathematics Level 2: Practice and Study Guide, GED Math: Quantitative, Arithmetic & Algebraic Problem Solving, CLEP College Algebra: Study Guide & Test Prep, Holt McDougal Algebra 2: Online Textbook Help, McDougal Littell Algebra 2: Online Textbook Help, Algebra II Curriculum Resource & Lesson Plans, Accuplacer Math: Quantitative Reasoning, Algebra, and Statistics Placement Test Study Guide, OSAT Middle Level/Intermediate Mathematics (CEOE) (125): Practice & Study Guide, Explorations in Core Math - Algebra 2: Online Textbook Help, Intermediate Algebra for College Students, NY Regents Exam - Algebra II: Test Prep & Practice, Create an account to start this course today. When simplifying math expressions, you can't simply proceed from left to right, multiplying, adding, subtracting, and so on as you go. Write answers with positive exponents. What Are the Five Main Exponent Properties? x(6 - x) can be simplified as 6x - x2, and -x(3 - x) can be simplified as -3x + x2. This is our simplified answer with positive exponents. Suppose you want the value y x. Choose "Simplify" from the topic selector and click to see the result in our Algebra Calculator! 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. Let's rewrite this with like terms over each other: 5/15 times x^2 / x^4 times y^9/y^9. For any real number [latex]a[/latex] and natural numbers [latex]m[/latex] and [latex]n[/latex], the product rule of exponents states that. Our expert tutors are available 24/7 to give you the answer you need in real-time. This will give us (8p)^3q^4 in the bottom or denominator, but our top or numerator will stay the same. Here is an example: 2x^2+x (4x+3) How to Use the Negative Simplify Solve - Properties of rational exponents calculator. To simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. Simplify expressions with positive exponents calculator - This Simplify expressions with positive exponents calculator helps to fast and easily solve any math. a1 n = na. 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. You need to provide a valid expression that involves exponents. Simplifies expressions step-by-step and shows the work! For example, 2x (x + y) can be simplified as 2x 2 + 2xy. Therefore, 3/4x + y/2 (4x + 7) = 3/4x + 2xy + 7y/2. This video looks at how to work with expressions that have rational exponents (fractions in the exponent). Get detailed solutions to your math problems with our Combining like terms step-by-step calculator. 986+ Experts. [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].