Let's solve the problem step-by-step:
Given the expression:
[tex]\[
\left[\left(\frac{2}{3}\right)^4\right]^5 \div \left[\left(\frac{3}{2}\right)^{-6} \cdot \left(\frac{2}{3}\right)^5\right]^2
\][/tex]
### Step 1: Simplify inside the parentheses
First, we deal with the expressions inside the parentheses.
[tex]\[
\left(\frac{2}{3}\right)^4
\][/tex]
[tex]\[
\left(\frac{3}{2}\right)^{-6} = \left(\frac{2}{3}\right)^6
\][/tex]
[tex]\[
\left(\frac{2}{3}\right)^5
\][/tex]
### Step 2: Apply the power to the exponentiated terms
For the term on the left:
[tex]\[
\left[\left(\frac{2}{3}\right)^4\right]^5 = \left(\frac{2}{3}\right)^{4 \times 5} = \left(\frac{2}{3}\right)^{20}
\][/tex]
For the term on the right, combine the terms inside the parentheses:
[tex]\[
\left[\left(\frac{3}{2}\right)^{-6} \cdot \left(\frac{2}{3}\right)^5\right] = \left(\frac{2}{3}\right)^6 \cdot \left(\frac{2}{3}\right)^5
\][/tex]
Adding the exponents:
[tex]\[
\left(\frac{2}{3}\right)^6 \cdot \left(\frac{2}{3}\right)^5 = \left(\frac{2}{3}\right)^{6+5} = \left(\frac{2}{3}\right)^{11}
\][/tex]
Now raise it to the power of 2:
[tex]\[
\left[\left(\frac{2}{3}\right)^{11}\right]^2 = \left(\frac{2}{3}\right)^{11 \times 2} = \left(\frac{2}{3}\right)^{22}
\][/tex]
### Step 3: Divide the terms
Now, divide the expressions:
[tex]\[
\left(\frac{2}{3}\right)^{20} \div \left(\frac{2}{3}\right)^{22} = \left(\frac{2}{3}\right)^{20-22} = \left(\frac{2}{3}\right)^{-2}
\][/tex]
When you have a negative exponent, you can take the reciprocal of the base and change the sign of the exponent:
[tex]\[
\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2
\][/tex]
Taking the square of the fraction:
[tex]\[
\left(\frac{3}{2}\right)^2 = \frac{3^2}{2^2} = \frac{9}{4} = 2.25
\][/tex]
Therefore:
[tex]\[
\left[\left(\frac{2}{3}\right)^4\right]^5 \div \left[\left(\frac{3}{2}\right)^{-6} \cdot \left(\frac{2}{3}\right)^5\right]^2 = 2.25
\][/tex]
