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Chapter 2: Problem 15

Simplify the given expression by writing it as a power of a single variable. $$ x^{5}\left(x^{2}\right)^{3} $$

Short Answer

Expert verified
The simplified expression as a power of a single variable is \(x^{11}\).

Step by step solution

01

Identify the base and exponents

Identify the base (x) and exponents (5, 2, 3) in the expression \(x^5(x^2)^3\).
02

Apply the rule for raising a power to another power

Apply the rule \((a^m)^n = a^{mn}\) to simplify the expression \((x^2)^3\). Multiply the exponents 2 and 3 to get the new exponent: \((x^2)^3 = x^{2 \times 3} = x^6\) Now, the expression becomes: \(x^5 \cdot x^6\)
03

Apply the rule for multiplying expressions with the same base

Apply the rule \(a^m * a^n = a^{m+n}\) to simplify the expression \(x^5 \cdot x^6\). Add the exponents 5 and 6 to get the new exponent: \(x^5 \cdot x^6 = x^{5+6} = x^{11}\) The simplified expression as a power of a single variable is \(x^{11}\).

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Most popular questions from this chapter

Find the asymptotes of the graph of the given function \(\mathrm{r}\). $$ r(x)=\frac{9 x+5}{x^{2}-x-6} $$

Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{2 x+1}{x-3} $$

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

Factor \(x^{8}-y^{8}\) as nicely as possible.

Suppose \(t\) is a zero of the polynomial \(p\) defined by $$ p(x)=3 x^{5}+7 x^{4}+2 x+6 $$ Show that \(\frac{1}{t}\) is a zero of the polynomial \(q\) defined by $$ q(x)=3+7 x+2 x^{4}+6 x^{5} $$.
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