Answer :
Answer:
frac{32}{19683}
Step-by-step explanation:
To determine the number by which \((\frac{2}{3})^3\) should be multiplied so that the quotient is \((\frac{27}{2})^2\), we can set up the following equation:
\[
\frac{(\frac{2}{3})^3}{x} = \left(\frac{27}{2}\right)^2
\]
First, simplify both sides of the equation:
\[
(\frac{2}{3})^3 = \frac{2^3}{3^3} = \frac{8}{27}
\]
\[
(\frac{27}{2})^2 = \frac{27^2}{2^2} = \frac{729}{4}
\]
So the equation becomes:
\[
\frac{\frac{8}{27}}{x} = \frac{729}{4}
\]
To isolate \( x \), multiply both sides by \( x \):
\[
\frac{8}{27} = x \cdot \frac{729}{4}
\]
Then, solve for \( x \) by dividing both sides by \(\frac{729}{4}\):
\[
x = \frac{\frac{8}{27}}{\frac{729}{4}} = \frac{8}{27} \cdot \frac{4}{729} = \frac{8 \times 4}{27 \times 729} = \frac{32}{19683}
\]
So, the number by which \((\frac{2}{3})^3\) should be multiplied is:
\[
\frac{32}{19683}
\]