Let's break down the given problem step by step.
### Step 1: Finding [tex]\( x \)[/tex]
Given:
[tex]\[ x = \left(\frac{2}{3}\right)^{-4} \times \left(\frac{3}{2}\right)^2 \][/tex]
We need to simplify this expression to find the value of [tex]\( x \)[/tex].
### Step 2: Simplifying [tex]\( x \)[/tex]
The expression involves two components with exponents. We can simplify each part separately.
First part:
[tex]\[ \left(\frac{2}{3}\right)^{-4} \][/tex]
Using the property of exponents, [tex]\( a^{-n} = \frac{1}{a^n} \)[/tex]:
[tex]\[ \left(\frac{2}{3}\right)^{-4} = \left(\frac{3}{2}\right)^4 = \left(\frac{3 \times 3 \times 3 \times 3}{2 \times 2 \times 2 \times 2}\right) = \frac{81}{16} \][/tex]
Second part:
[tex]\[ \left(\frac{3}{2}\right)^2 \][/tex]
[tex]\[ \left(\frac{3}{2}\right)^2 = \frac{3 \times 3}{2 \times 2} = \frac{9}{4} \][/tex]
Now multiply these two results:
[tex]\[ x = \frac{81}{16} \times \frac{9}{4} \][/tex]
Combine the fractions by multiplying the numerators and the denominators:
[tex]\[ x = \frac{81 \times 9}{16 \times 4} = \frac{729}{64} \][/tex]
So, we have:
[tex]\[ x = \frac{729}{64} \][/tex]
This can be approximated:
[tex]\[ x \approx 11.390625 \][/tex]
### Step 3: Finding [tex]\( (x)^{-2} \)[/tex]
Next, let's find [tex]\( (x)^{-2} \)[/tex].
Using [tex]\( x \approx 11.390625 \)[/tex]:
[tex]\[ (x)^{-2} = (11.390625)^{-2} \][/tex]
This can be approximated to:
[tex]\[ (11.390625)^{-2} \approx 0.007707 \][/tex]
### Step 4: Finding [tex]\( (x)^{-1} \)[/tex]
Finally, let's find [tex]\( (x)^{-1} \)[/tex].
Using [tex]\( x \approx 11.390625 \)[/tex]:
[tex]\[ (x)^{-1} = (11.390625)^{-1} \][/tex]
This approximates to:
[tex]\[ (11.390625)^{-1} \approx 0.087791 \][/tex]
### Summary
The values are:
(a) [tex]\( (x)^{-2} = 0.007707 \)[/tex]
(b) [tex]\( (x)^{-1} = 0.087791 \)[/tex]
These values match the given numerical results.
