To simplify the given expression [tex]\(\left(\frac{2 x^{-3}}{5 w^2}\right)^{-3}\)[/tex] using only positive exponents, follow these steps:
### Step-by-Step Solution:
1. Rewrite the Expression Inside the Parentheses:
[tex]\[
\frac{2 x^{-3}}{5 w^2}
\][/tex]
This is already in a form that's manageable.
2. Apply the Negative Exponent to the Entire Fraction:
[tex]\[
\left(\frac{2 x^{-3}}{5 w^2}\right)^{-3}
\][/tex]
Recall that [tex]\((a/b)^{-n} = (b/a)^n\)[/tex]. Apply this property:
[tex]\[
\left(\frac{2 x^{-3}}{5 w^2}\right)^{-3} = \left(\frac{5 w^2}{2 x^{-3}}\right)^{3}
\][/tex]
3. Simplify the Fraction Inside the Parentheses:
[tex]\[
\frac{5 w^2}{2 x^{-3}}
\][/tex]
Recall that [tex]\(x^{-3} = \frac{1}{x^3}\)[/tex], so rewrite it:
[tex]\[
\frac{5 w^2}{2 \cdot \frac{1}{x^3}} = \frac{5 w^2 \cdot x^3}{2}
\][/tex]
Which simplifies to:
[tex]\[
\frac{5 w^2 x^3}{2}
\][/tex]
4. Raise the Simplified Fraction to the Power of 3:
Now raise the simplified fraction to the power of 3:
[tex]\[
\left(\frac{5 w^2 x^3}{2}\right)^3
\][/tex]
Apply the power rule [tex]\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)[/tex]:
[tex]\[
\frac{(5 w^2 x^3)^3}{2^3}
\][/tex]
5. Simplify the Numerator and Denominator Separately:
- For the numerator:
[tex]\[
(5 w^2 x^3)^3 = 5^3 (w^2)^3 (x^3)^3 = 125 w^6 x^9
\][/tex]
- For the denominator:
[tex]\[
2^3 = 8
\][/tex]
6. Combine the Results:
[tex]\[
\frac{125 w^6 x^9}{8}
\][/tex]
### Final Answer:
[tex]\[
\boxed{\frac{125 w^6 x^9}{8}}
\][/tex]
