Certainly! Let's simplify the given expressions step-by-step using the reciprocal rule for negative exponents.
Expression 1:
[tex]\[
\left(\frac{2}{5}\right)^{-2}
\][/tex]
Step 1: Apply the reciprocal rule, which states that [tex]\(\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n\)[/tex].
[tex]\[
\left(\frac{2}{5}\right)^{-2} = \left(\frac{5}{2}\right)^2
\][/tex]
Step 2: Raise the fraction [tex]\(\left(\frac{5}{2}\right)\)[/tex] to the power of [tex]\(2\)[/tex].
[tex]\[
\left(\frac{5}{2}\right)^2 = \frac{5^2}{2^2} = \frac{25}{4} = 6.25
\][/tex]
So, the simplified form of [tex]\(\left(\frac{2}{5}\right)^{-2}\)[/tex] is:
[tex]\[
6.25
\][/tex]
Expression 2:
[tex]\[
\left(\frac{x^3}{7}\right)^{-3}
\][/tex]
Step 1: Apply the reciprocal rule, which states that [tex]\(\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n\)[/tex].
[tex]\[
\left(\frac{x^3}{7}\right)^{-3} = \left(\frac{7}{x^3}\right)^3
\][/tex]
Step 2: Raise the fraction [tex]\(\left(\frac{7}{x^3}\right)\)[/tex] to the power of [tex]\(3\)[/tex].
[tex]\[
\left(\frac{7}{x^3}\right)^3 = \frac{7^3}{(x^3)^3} = \frac{343}{x^9}
\][/tex]
So, the simplified form of [tex]\(\left(\frac{x^3}{7}\right)^{-3}\)[/tex] is:
[tex]\[
\frac{343}{x^9}
\][/tex]
Thus, the final simplified answers are:
[tex]\[
\left(\frac{2}{5}\right)^{-2} = 6.25
\][/tex]
[tex]\[
\left(\frac{x^3}{7}\right)^{-3} = \frac{343}{x^9}
\][/tex]
