Question

#6 spicy! which expression is equivalent to \\(\frac{(5x^7)^{-3}}{x^5}\\) for all values of x where the expression is defined?
a. \\(\frac{1}{125x^{27}}\\)
b. \\(\frac{1}{125x^{26}}\\)
c. \\(\frac{1}{25x^{27}}\\)
d. \\(\frac{1}{25x^{26}}\\)

Explanation:

Step1: Expand the numerator's power

First, apply the power of a product rule: $(ab)^n=a^n b^n$, then the power rule: $(x^m)^n=x^{m \cdot n}$.
$$(5x^7)^{-3}=5^{-3} \cdot (x^7)^{-3} = \frac{1}{5^3} \cdot x^{-21} = \frac{1}{125x^{21}}$$

Step2: Rewrite the entire expression

Substitute the expanded numerator into the original fraction, then use the rule $\frac{1/x^m}{x^n}=\frac{1}{x^{m+n}}$.
$$\frac{(5x^7)^{-3}}{x^5} = \frac{1}{125x^{21} \cdot x^5} = \frac{1}{125x^{21+5}}$$

Step3: Simplify the exponent

Add the exponents of $x$.
$$\frac{1}{125x^{26}}$$

Answer:

b. $\frac{1}{125x^{26}}$