Question

2 over 3 to the power of 4 times 2 over 3 to the power of -6

Answer 1

To solve the expression \( \left( \frac{2}{3} \right)^4 \times \left( \frac{2}{3} \right)^{-6} \), we can use the property of exponents that states \( a^m \times a^n = a^{m+n} \).

Here, we have:

\[
\left( \frac{2}{3} \right)^4 \times \left( \frac{2}{3} \right)^{-6} = \left( \frac{2}{3} \right)^{4 + (-6)}
\]

Calculating the exponent:

\[
4 + (-6) = 4 - 6 = -2
\]

Now we can rewrite the expression:

\[
\left( \frac{2}{3} \right)^{-2}
\]

Using the property \( a^{-m} = \frac{1}{a^m} \):

\[
\left( \frac{2}{3} \right)^{-2} = \frac{1}{\left( \frac{2}{3} \right)^2} = \frac{1}{\frac{2^2}{3^2}} = \frac{1}{\frac{4}{9}} = \frac{9}{4}
\]

Thus, the final answer is:

\[
\frac{9}{4}
\]