Sure, let's simplify the expression [tex]\(\frac{2 x^3 y^{-2}}{3 z^{-2}}\)[/tex] step by step.
1. Understand the expression:
- The numerator is [tex]\(2 x^3 y^{-2}\)[/tex].
- The denominator is [tex]\(3 z^{-2}\)[/tex].
2. Rewrite the negative exponents to positive exponents:
- Recall that [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex], so [tex]\(y^{-2}\)[/tex] becomes [tex]\(\frac{1}{y^2}\)[/tex] and [tex]\(z^{-2}\)[/tex] becomes [tex]\(\frac{1}{z^2}\)[/tex].
- Substitute these into the expression to get:
[tex]\[
\frac{2 x^3 \cdot \frac{1}{y^2}}{3 \cdot \frac{1}{z^2}}
\][/tex]
3. Simplify the fraction:
- The expression can be simplified by multiplying by the reciprocal of the denominator:
[tex]\[
\frac{2 x^3 \cdot \frac{1}{y^2}}{3 \cdot \frac{1}{z^2}} = \frac{2 x^3 \cdot \frac{1}{y^2}}{\frac{3}{z^2}}
\][/tex]
- This is the same as:
[tex]\[
\frac{2 x^3}{3} \cdot \frac{z^2}{y^2}
\][/tex]
4. Combine the fractions:
- Multiply the terms in the numerator and denominator:
[tex]\[
\frac{2 x^3 \cdot z^2}{3 y^2}
\][/tex]
5. Final expression:
- The fraction is now fully simplified. The simplified form of [tex]\(\frac{2 x^3 y^{-2}}{3 z^{-2}}\)[/tex] is:
[tex]\[
\frac{2 x^3 z^2}{3 y^2}
\][/tex]
So, the simplified expression is [tex]\(\boxed{\frac{2 x^3 z^2}{3 y^2}}\)[/tex].
