Let's evaluate the expression step-by-step.
### Expression to Evaluate
[tex]\[
\left[\left(12^2 + 5^2\right)^{\frac{1}{2}}\right]^3
\][/tex]
### Step 1: Calculate the squares inside the parentheses
First, we'll square the individual numbers inside the parentheses:
[tex]\[
12^2 = 144
\][/tex]
[tex]\[
5^2 = 25
\][/tex]
### Step 2: Add the results of the squares
Next, we add these two squared values together:
[tex]\[
12^2 + 5^2 = 144 + 25 = 169
\][/tex]
### Step 3: Calculate the square root of the sum
Now we find the square root of the result obtained in the previous step (\(169\)):
[tex]\[
\left(169\right)^{\frac{1}{2}} = \sqrt{169} = 13
\][/tex]
### Step 4: Raise the square root to the power of 3
Finally, we raise the result (which is 13) to the power of 3:
[tex]\[
13^3 = 13 \times 13 \times 13 = 2197
\][/tex]
### Final Result
By following these steps, we find that:
[tex]\[
\left[\left(12^2 + 5^2\right)^{\frac{1}{2}}\right]^3 = 2197
\][/tex]
Therefore, the evaluated result of the given expression is:
[tex]\[
2197
\][/tex]
