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[tex]$\left(5,5^3 \cdot 5^2\right)^2:\left[(5)^3\right]^2=$[/tex]
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Response:
Simplify the following expression:
[tex]\[ \left(5,5^3 \cdot 5^2\right)^2 : \left[(5)^3\right]^2 \][/tex]



Answer :

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]