Answer :

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]