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Chapter 2: Problem 20

Simplify the given expression by writing it as a power of a single variable. $$ w^{3}\left(w^{4}\left(w^{-3}\right)^{6}\right)^{2} $$

Short Answer

Expert verified
The simplified expression is \( w^{-25} \).

Step by step solution

01

Expand the expression inside the parentheses

To expand the expression inside the parentheses, we will raise each term to the power of 6: \( w^{-3} \) raised to the power of 6 will be using rule \( (a^m)^n = a^{mn} \) \( (w^{-3})^{6} = w^{-18} \). Now, the expression becomes: \( w^3(w^4w^{-18})^2 \).
02

Simplify the expression inside the inner parentheses

Inside the inner parentheses, we have \(w^4w^{-18}\). We can simplify this expression using the rule \( a^{m} \cdot a^{n} = a^{m + n} \) since both have the same base (w): \( w^4w^{-18} = w^{4 - 18} = w^{-14} \). The expression now becomes: \( w^3(w^{-14})^2 \).
03

Raise the expression inside the parentheses to the power of 2

We now apply the rule \( (a^m)^n = a^{mn} \) to raise the expression inside the parentheses to the power of 2: \( (w^{-14})^2 = w^{-28} \). The expression now becomes: \( w^3w^{-28} \).
04

Simplify the expression

Finally, we will simplify the expression using the rule \( a^{m} \cdot a^{n} = a^{m + n} \) for expressions with the same base: \( w^3w^{-28} = w^{3 - 28} = w^{-25} \). So the simplified expression is: \( w^{-25} \).

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Most popular questions from this chapter

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (s(x))^{2} t(x) $$

Give an example of polynomials \(p\) and \(q\) of degree 3 such that \(p(1)=q(1), p(2)=q(2),\) and \(p(3)=q(3),\) but \(p(4) \neq q(4)\).

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (3 r-2 s)(x) $$

Suppose \(p\) and \(q\) are polynomials of degree 3 such that \(p(1)=q(1), p(2)=q(2), p(3)=q(3),\) and \(p(4)=q(4) .\) Explain why \(p=q\).

Find all real numbers \(x\) such that $$ x^{6}-8 x^{3}+15=0 $$.
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