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Chapter 2: Problem 32

Simplify. \(\left(6 s^{8} t^{3}\right)^{2}\left(-\frac{10}{3} s t^{4}\right)^{2}\)

Short Answer

Expert verified
The simplified expression is \(\boxed{400s^{18}t^{14}}\).

Step by step solution

01

Apply the Exponent Rules

First, let's apply the power rule that states that \((a^n)^m = a^{nm}\) and the product rule that states that \(a^n\cdot a^m = a^{n+m}\) for any base \(a\), and positive integer exponents \(m\) and \(n\). Note that since the terms are inside the parentheses, we will apply the exponent to each term separately. \[\begin{aligned} \left(6 s^{8} t^{3}\right)^{2}\left(-\frac{10}{3} s t^{4}\right)^{2} &= (6^2)(s^{8\cdot2})(t^{3\cdot2})(-\frac{10}{3})^2(s^2)(t^{4\cdot2}) \\ \end{aligned}\]
02

Calculate the Exponents

Now we have to calculate the exponents and the coefficients separately: \[\begin{aligned} &= (36)(s^{16})(t^{6})(\frac{100}{9})(s^2)(t^{8}) \\ \end{aligned}\]
03

Multiply Same Bases and Add Their Exponents

Next, we will multiply the terms with the same bases (i.e., \(s\) and \(t\)) and add their exponents together. We also multiply their coefficients: \[\begin{aligned} &= \frac{3600}{9}(s^{16+2})(t^{6+8}) \\ \end{aligned}\]
04

Simplify the Exponents and Coefficients

To complete the simplification, let's add the exponents of \(s\) and \(t\) and simplify the coefficient: \[\begin{aligned} &= \frac{3600}{9}(s^{18})(t^{14}) \\ \end{aligned}\]
05

Final Answer

Now, we have the simplified form of the given expression: \[\boxed{400s^{18}t^{14}}\]

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Most popular questions from this chapter

Simplify the expression using one of the power rules. $$-8(m n)^{5}$$

Simplify the expression using the product rule. Leave your answer in exponential form. $$\left(12 c^{3}\right)\left(\frac{14}{15} c^{2}\right)\left(\frac{5}{7} c^{6}\right)$$

Simplify the expression using the product rule. Leave your answer in exponential form.

Simplify using the quotient rule. $$\frac{5 m^{-1} n^{-6}}{15 m^{-5} n^{2}}$$

Perform the operation as indicated. Write the final answer without an exponent. \(\frac{-4.5 \times 10^{-6}}{-1.5 \times 10^{-8}}\)
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