To solve for [tex]\(\left(\frac{q}{p}\right)^3\)[/tex] given that [tex]\(\frac{p}{q} = \left(\frac{2}{3}\right)^2 \div \left(\frac{6}{7}\right)^0\)[/tex], follow these steps:
1. Simplify the Right-Hand Side Expression:
- [tex]\(\left(\frac{2}{3}\right)^2\)[/tex] is evaluated first. This gives:
[tex]\[
\left(\frac{2}{3}\right)^2 = \frac{4}{9}
\][/tex]
- [tex]\(\left(\frac{6}{7}\right)^0\)[/tex] is a number raised to the power of 0. Any non-zero number raised to the power of 0 is 1:
[tex]\[
\left(\frac{6}{7}\right)^0 = 1
\][/tex]
2. Evaluate the Division:
- Now, we have:
[tex]\[
\frac{p}{q} = \frac{4}{9} \div 1
\][/tex]
- Since dividing by 1 leaves the number unchanged:
[tex]\[
\frac{p}{q} = \frac{4}{9}
\][/tex]
3. Determine [tex]\(\frac{q}{p}\)[/tex]:
- The reciprocal of [tex]\(\frac{p}{q}\)[/tex] will give us [tex]\(\frac{q}{p}\)[/tex]:
[tex]\[
\frac{q}{p} = \frac{1}{\frac{4}{9}} = \frac{9}{4}
\][/tex]
4. Calculate [tex]\(\left(\frac{q}{p}\right)^3\)[/tex]:
- Raise [tex]\(\frac{q}{p}\)[/tex] to the power of 3:
[tex]\[
\left(\frac{q}{p}\right)^3 = \left(\frac{9}{4}\right)^3
\][/tex]
- Evaluate [tex]\(\left(\frac{9}{4}\right)^3\)[/tex]:
[tex]\[
\left(\frac{9}{4}\right)^3 = \frac{9^3}{4^3} = \frac{729}{64}
\][/tex]
5. Interpreting the Decimal Form Result:
- When simplified to a decimal form, [tex]\(\frac{729}{64} \approx 11.390625\)[/tex].
Therefore, [tex]\(\left(\frac{q}{p}\right)^3 = 11.390625\)[/tex].
