Sure, let's break down the problem step by step to find the value of [tex]\(x\)[/tex] and then [tex]\(x^{-2}\)[/tex].
Given the expression for [tex]\(x\)[/tex]:
[tex]\[ x = \left(\frac{5}{2}\right)^2 \times \left(\frac{2}{5}\right)^{-3} \][/tex]
First, let's simplify each part of the expression.
1. Evaluate [tex]\(\left(\frac{5}{2}\right)^2\)[/tex]:
[tex]\[ \left(\frac{5}{2}\right)^2 = \frac{5^2}{2^2} = \frac{25}{4} \][/tex]
2. Evaluate [tex]\(\left(\frac{2}{5}\right)^{-3}\)[/tex]:
[tex]\[ \left(\frac{2}{5}\right)^{-3} = \left(\frac{5}{2}\right)^3 = \frac{5^3}{2^3} = \frac{125}{8} \][/tex]
Next, we multiply these results together:
[tex]\[ x = \left(\frac{25}{4}\right) \times \left(\frac{125}{8}\right) = \frac{25 \times 125}{4 \times 8} = \frac{3125}{32} \][/tex]
The value of [tex]\(x\)[/tex] is:
[tex]\[ x \approx 97.65625 \][/tex]
Now, let's find [tex]\(x^{-2}\)[/tex]:
[tex]\[ x^{-2} = \left(\frac{3125}{32}\right)^{-2} = \left(\frac{32}{3125}\right)^2 = \frac{32^2}{3125^2} = \frac{1024}{9765625} \][/tex]
Which simplifies to:
[tex]\[ x^{-2} \approx 0.00010485760000000003 \][/tex]
Thus, the values are:
[tex]\[ x \approx 97.65625 \][/tex]
[tex]\[ x^{-2} \approx 0.00010485760000000003 \][/tex]
So, the value of [tex]\(x^{-2}\)[/tex] is approximately [tex]\(0.00010485760000000003\)[/tex].
