Question
Simplify this expression: `\left(4x^{3}y^{-4}\right)^{-2}`.
Answers
Answered by
shmunklee
Simplify this expression: `\left(20a^{2}b^{-12}c^{-4}\right)\left(3a^{15}b^{12}c^{6}\right)`.
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say it in a easier way
Answered by
GPT-4o mini
To simplify the expression \(\left(4x^{3}y^{-4}\right)^{-2}\), we will apply the property of exponents that states \((a^m)^{-n} = a^{-mn}\) and the rule that states \((ab)^n = a^n b^n\).
1. Distribute the exponent \(-2\) to each factor inside the parentheses:
\[
\left(4x^{3}y^{-4}\right)^{-2} = 4^{-2} \cdot (x^{3})^{-2} \cdot (y^{-4})^{-2}
\]
2. Now simplify each part:
- \(4^{-2} = \frac{1}{4^2} = \frac{1}{16}\)
- \((x^{3})^{-2} = x^{-6}\)
- \((y^{-4})^{-2} = y^{8}\) (since \(-4 \cdot -2 = 8\))
Putting these together, we have:
\[
\left(4x^{3}y^{-4}\right)^{-2} = \frac{1}{16} \cdot x^{-6} \cdot y^{8}
\]
3. Finally, rewrite the expression in a more standard form:
\[
\frac{y^{8}}{16x^{6}}
\]
Thus, the simplified expression is:
\[
\frac{y^{8}}{16x^{6}}
\]
1. Distribute the exponent \(-2\) to each factor inside the parentheses:
\[
\left(4x^{3}y^{-4}\right)^{-2} = 4^{-2} \cdot (x^{3})^{-2} \cdot (y^{-4})^{-2}
\]
2. Now simplify each part:
- \(4^{-2} = \frac{1}{4^2} = \frac{1}{16}\)
- \((x^{3})^{-2} = x^{-6}\)
- \((y^{-4})^{-2} = y^{8}\) (since \(-4 \cdot -2 = 8\))
Putting these together, we have:
\[
\left(4x^{3}y^{-4}\right)^{-2} = \frac{1}{16} \cdot x^{-6} \cdot y^{8}
\]
3. Finally, rewrite the expression in a more standard form:
\[
\frac{y^{8}}{16x^{6}}
\]
Thus, the simplified expression is:
\[
\frac{y^{8}}{16x^{6}}
\]