Let's solve the given expression step-by-step:
Given expression: \(\left( 5, 5^3 \cdot 5^2 \right)^2 : \left[(5)^3\right]^2 \)
### Step 1: Simplify Inside the Parentheses
First, let's simplify the term inside the parentheses:
[tex]\[ 5^3 \cdot 5^2 \][/tex]
Since the bases are the same, we can add the exponents:
[tex]\[ 5^3 \cdot 5^2 = 5^{3+2} = 5^5 \][/tex]
So the expression inside the parentheses becomes:
[tex]\[ \left( 5, 5^5 \right)^2 \][/tex]
Considering this numerical value:
[tex]\[ 5^5 = 3125 \][/tex]
### Step 2: Evaluate the Numerator
Next, let's consider the expression \(\left( 5, 3125 \right)^2\). We need to square this entire term:
[tex]\[ (5, 3125)^2 \][/tex]
This means squaring both elements in turn:
[tex]\[ (5^2, 3125^2) \][/tex]
We know:
[tex]\[ 5^2 = 25 \][/tex]
[tex]\[ 3125^2 = 9765625 \][/tex]
So the square of the numerator is:
[tex]\[ (25, 9765625) \][/tex]
### Step 3: Evaluate the Denominator
Now, let's consider the denominator:
[tex]\[ \left[(5)^3\right]^2 \][/tex]
This can be rewritten as:
[tex]\[ (5^3)^2 \][/tex]
Using the power rule \((a^m)^n = a^{mn}\), we get:
[tex]\[ (5^3)^2 = 5^{3 \cdot 2} = 5^6 \][/tex]
Now, evaluating the numerical value:
[tex]\[ 5^6 = 15625 \][/tex]
### Step 4: Simplify the Entire Expression
We now have:
[tex]\[ \left( 25 \cdot 9765625 \right) : 15625 \][/tex]
First, multiply the terms inside the numerator:
[tex]\[ 25 \cdot 9765625 = 244140625 \][/tex]
So we need to divide:
[tex]\[ \frac{244140625}{15625} = 15625 \][/tex]
### Step 5: Final Answer
So, the final result of the given expression is:
[tex]\[ \boxed{15625} \][/tex]
