Question

Which is the simplified form of the expression (StartFraction (x Superscript negative 3 Baseline) (y squared) Over (x Superscript 4 Baseline) (y superscript 6 Baseline) EndFraction) cubed?
StartFraction 1 Over x Superscript 4 Baseline y EndFraction
StartFraction 1 Over x Superscript 21 Baseline y Superscript 12 Baseline EndFraction
x cubed y Superscript 12
x cubed y Superscript 24

Answer 1

To simplify the expression \(\left(\frac{x^{-3}y^2}{x^4y^6}\right)^3\), we can first simplify the fraction inside the parentheses.

1. Simplify the fraction:
\[
\frac{x^{-3}y^2}{x^4y^6} = \frac{y^2}{y^6} \cdot \frac{x^{-3}}{x^4} = \frac{y^{2-6}}{x^{3+4}} = \frac{y^{-4}}{x^{7}} = \frac{1}{x^{7}y^{4}}
\]

2. Now, we need to cube the simplified fraction:
\[
\left(\frac{1}{x^7 y^4}\right)^3 = \frac{1^3}{(x^7)^3(y^4)^3} = \frac{1}{x^{21}y^{12}}
\]

So, the simplified form of the expression is:
\[
\frac{1}{x^{21}y^{12}}
\]

Therefore, the answer is:
\(\mathbf{\frac{1}{x^{21} y^{12}}}\)