Question

simplify using the laws of exponents.
$25^{\frac{-3}{2}}$
\\(\circ\\) -125
\\(\circ\\) -37.5
\\(\circ\\) \\(\frac{1}{125}\\)
\\(\circ\\) \\(\frac{1}{15}\\)

Explanation:

Step1: Recall negative exponent rule

The negative exponent rule states that \( a^{-n} = \frac{1}{a^n} \). So, we can rewrite \( 25^{-\frac{3}{2}} \) as \( \frac{1}{25^{\frac{3}{2}}} \).

Step2: Recall fractional exponent rule

The fractional exponent rule is \( a^{\frac{m}{n}}=\sqrt[n]{a^m} \) or \( (\sqrt[n]{a})^m \). For \( 25^{\frac{3}{2}} \), we can first take the square root of 25 (since the denominator of the fraction is 2) and then raise it to the power of 3 (since the numerator of the fraction is 3).

The square root of 25 is 5, so \( 25^{\frac{3}{2}}=(\sqrt{25})^3 = 5^3 \).

Step3: Calculate \( 5^3 \)

\( 5^3 = 5\times5\times5 = 125 \).

Step4: Substitute back

We had \( \frac{1}{25^{\frac{3}{2}}} \), and since \( 25^{\frac{3}{2}} = 125 \), this becomes \( \frac{1}{125} \).

Answer:

\(\frac{1}{125}\)