`(5xx5xx5xx5xx5xx5)^(4)xx(5xx5)^(6)div(5)^(2)=(25)^(?)`

Options

A. 10
B. 17
C. 19
D. 12

Explanation

To solve the equation \((5 \times 5 \times 5 \times 5 \times 5)^{4} \times (5 \times 5)^{6} \div (5)^{2} = (25)^{?}\), we will simplify each part step by step.

### Step 1: Rewrite the expression using exponents
We know that:
- \(5 \times 5 = 5^2\)
- \(5 \times 5 \times 5 = 5^3\)
- \(5 \times 5 \times 5 \times 5 = 5^4\)

Thus, we can rewrite the expression as:
\[ (5^5)^{4} \times (5^2)^{6} \div (5^2) \]

### Step 2: Apply the power of a power property
Using the property \((a^m)^n = a^{m \cdot n}\), we simplify:
\[ (5^5)^{4} = 5^{5 \cdot 4} = 5^{20} \]
\[ (5^2)^{6} = 5^{2 \cdot 6} = 5^{12} \]

Now our expression looks like:
\[ 5^{20} \times 5^{12} \div 5^{2} \]

### Step 3: Combine the terms using the product of powers property
Using the property \(a^m \times a^n = a^{m+n}\), we combine the powers:
\[ 5^{20} \times 5^{12} = 5^{20 + 12} = 5^{32} \]

### Step 4: Apply the division property of exponents
Using the property \(\frac{a^m}{a^n} = a^{m-n}\), we simplify:
\[ \frac{5^{32}}{5^{2}} = 5^{32 - 2} = 5^{30} \]

### Step 5: Rewrite \(25\) as a power of \(5\)
We know that \(25 = 5^2\). Therefore:
\[ (25)^{?} = (5^2)^{?} = 5^{2?} \]

### Step 6: Set the exponents equal to each other
Now we have:
\[ 5^{30} = 5^{2?} \]
This implies:
\[ 30 = 2? \]

### Step 7: Solve for \(?\)
To find \(?\), divide both sides by \(2\):
\[ ? = \frac{30}{2} = 15 \]

Thus, the final answer is:
\[ ? = 15 \]