To solve the given expression:
[tex]\[
\left[\left\{\left(\frac{5}{7}\right)^2\right\}^{-1}\right]^{-3}
\][/tex]
we will go through the following steps:
### Step 1: Simplify the inner-most expression
The inner-most expression is:
[tex]\[
\left(\frac{5}{7}\right)^2
\][/tex]
Calculate [tex]\(\left(\frac{5}{7}\right)^2\)[/tex]:
[tex]\[
\left(\frac{5}{7}\right)^2 = 0.5102040816326531
\][/tex]
### Step 2: Apply the exponent of [tex]\(-1\)[/tex]
Next, we take the result of [tex]\(\left(\frac{5}{7}\right)^2\)[/tex] and raise it to the power of [tex]\(-1\)[/tex]:
[tex]\[
\left[\left(\frac{5}{7}\right)^2\right]^{-1}
\][/tex]
This is equivalent to taking the reciprocal of [tex]\(0.5102040816326531\)[/tex], which gives us [tex]\(1.96\)[/tex]:
[tex]\[
0.5102040816326531^{-1} = 1.96
\][/tex]
### Step 3: Apply the exponent of [tex]\(-3\)[/tex]
Next, we take the result from Step 2 and raise it to the power of [tex]\(-3\)[/tex]:
[tex]\[
\left[1.96\right]^{-3}
\][/tex]
This simplifies and is calculated as follows:
[tex]\[
1.96^{-3} = 0.13281030862990761
\][/tex]
### Step 4: Combine the exponents
To simplify the entire expression in one step using the property of exponents: [tex]\(\left[\left(a^m\right)^n\right]^p = a^{m \cdot n \cdot p}\)[/tex]
So,
[tex]\[
\left[\left\{\left(\frac{5}{7}\right)^2\right\}^{-1}\right]^{-3} = \left(\frac{5}{7}\right)^{2 \cdot (-1) \cdot (-3)}
\][/tex]
The combined exponent is:
[tex]\[
2 \cdot (-1) \cdot (-3) = 6
\][/tex]
Thus, the expression simplifies to:
[tex]\[
\left(\frac{5}{7}\right)^{6}
\][/tex]
### Final Result
The simplified form of the given expression is:
[tex]\[
\left(\frac{5}{7}\right)^6
\][/tex]
The numerical value of this, as calculated, is:
[tex]\[
0.13281030862990761
\][/tex]
So, the fully simplified and numerically evaluated form of the given expression [tex]\(\left[\left\{\left(\frac{5}{7}\right)^2\right\}^{-1}\right]^{-3}\)[/tex] is:
[tex]\[
\left(\frac{5}{7}\right)^6 = 0.13281030862990761
\][/tex]
