Certainly! Let's solve this step-by-step.
Given:
[tex]\[ x = \left(\frac{2}{3}\right)^{-4} \times \left(\frac{3}{2}\right)^2 \][/tex]
### Step 1: Evaluate [tex]\( \left( \frac{2}{3} \right)^{-4} \)[/tex]
Using properties of exponents, [tex]\( \left( \frac{a}{b} \right)^{-n} = \left( \frac{b}{a} \right)^n \)[/tex]:
[tex]\[ \left( \frac{2}{3} \right)^{-4} = \left( \frac{3}{2} \right)^4 \][/tex]
### Step 2: Evaluate [tex]\( \left( \frac{3}{2} \right)^4 \)[/tex]
[tex]\[ \left( \frac{3}{2} \right)^4 = \left( \frac{3 \times 3 \times 3 \times 3}{2 \times 2 \times 2 \times 2} \right) = \frac{81}{16} \][/tex]
### Step 3: Evaluate [tex]\( \left( \frac{3}{2} \right)^2 \)[/tex]
[tex]\[ \left( \frac{3}{2} \right)^2 = \left( \frac{3 \times 3}{2 \times 2} \right) = \frac{9}{4} \][/tex]
### Step 4: Multiply the Results of Steps 2 and 3
[tex]\[ x = \left( \frac{3}{2} \right)^4 \times \left( \frac{3}{2} \right)^2 = \frac{81}{16} \times \frac{9}{4} \][/tex]
### Step 5: Simplify the Multiplication
[tex]\[ x = \frac{81 \times 9}{16 \times 4} = \frac{729}{64} \][/tex]
Given this interpretation and relaying the necessary calculations, the value of [tex]\( x \)[/tex] is found as follows:
(a) Calculate [tex]\( x^{-2} \)[/tex]
[tex]\[ x^{-2} = \left( \frac{729}{64} \right)^{-2} \][/tex]
Using the given answer:
[tex]\[ x^{-2} \approx 0.007707346629258937 \][/tex]
(b) The value of [tex]\( x \)[/tex]
Given the previous value calculated:
[tex]\[ x \approx 11.390625 \][/tex]
Thus, the final results are:
(a) [tex]\( x^{-2} \approx 0.007707346629258937 \)[/tex]
(b) [tex]\( x \approx 11.390625 \)[/tex]
