Let's solve the problem step-by-step:
First, we are given the expression for [tex]\( x \)[/tex]:
[tex]\[ x = \left( \frac{2}{3} \right)^{-4} \times \left( \frac{3}{2} \right)^2 \][/tex]
1. Evaluate [tex]\(\left( \frac{2}{3} \right)^{-4}\)[/tex]:
Recall that raising a fraction to a negative exponent involves taking the reciprocal of the fraction and then raising it to the positive value of the exponent. So,
[tex]\[
\left( \frac{2}{3} \right)^{-4} = \left( \frac{3}{2} \right)^4
\][/tex]
Now calculate [tex]\(\left( \frac{3}{2} \right)^4\)[/tex]:
[tex]\[
\left( \frac{3}{2} \right)^4 = \frac{3^4}{2^4} = \frac{81}{16}
\][/tex]
2. Evaluate [tex]\(\left( \frac{3}{2} \right)^2\)[/tex]:
Similarly,
[tex]\[
\left( \frac{3}{2} \right)^2 = \frac{3^2}{2^2} = \frac{9}{4}
\][/tex]
3. Multiply the evaluated expressions:
We need to multiply [tex]\( \frac{81}{16} \)[/tex] by [tex]\( \frac{9}{4} \)[/tex]:
[tex]\[
x = \frac{81}{16} \times \frac{9}{4} = \frac{81 \times 9}{16 \times 4} = \frac{729}{64}
\][/tex]
Therefore,
[tex]\[
x = 11.390625
\][/tex]
4. Find the value of [tex]\( b \cdot [x]^{-1} \)[/tex]:
We need to compute [tex]\([x]^{-1}\)[/tex], which is the reciprocal of [tex]\( x \)[/tex]:
[tex]\[
[x]^{-1} = \frac{1}{x} = \frac{1}{11.390625} = 0.08779149519890259
\][/tex]
Given [tex]\( b \)[/tex] (assuming [tex]\( b = 1 \)[/tex] for this specific solution), the final value is:
[tex]\[
b \cdot [x]^{-1} = 1 \cdot 0.08779149519890259 = 0.08779149519890259
\][/tex]
Hence, [tex]\( b \cdot [x]^{-1} = 0.08779149519890259 \)[/tex].
