To simplify the expression \(\frac{(x^{25})^{-6}}{(x^{-3})^{48}}\), we can follow these steps:
1. Apply the power rule, which states that \((x^a)^b = x^{a \cdot b}\), to simplify the exponents in the numerator and denominator.
- For the numerator \((x^{25})^{-6}\):
[tex]\[
(x^{25})^{-6} = x^{25 \cdot (-6)} = x^{-150}
\][/tex]
- For the denominator \((x^{-3})^{48}\):
[tex]\[
(x^{-3})^{48} = x^{-3 \cdot 48} = x^{-144}
\][/tex]
2. Next, express the original fraction with these simplified exponents:
[tex]\[
\frac{x^{-150}}{x^{-144}}
\][/tex]
3. When dividing exponents with the same base, we subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
x^{-150} / x^{-144} = x^{-150 - (-144)} = x^{-150 + 144} = x^{-6}
\][/tex]
Therefore, the power of [tex]\(x\)[/tex] in the simplified expression is [tex]\(-6\)[/tex].
