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

Options

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

Explanation

To solve the equation \((5 \times 5 \times 5 \times 5 \times 5 \times 5)^4 \times (5 \times 5)^6 \div (5)^2 = (25)^{?}\), we will follow these steps:

### Step 1: Rewrite the expression using exponents
The expression can be rewritten using exponents:
- \(5 \times 5 \times 5 \times 5 \times 5 \times 5 = 5^6\)
- \(5 \times 5 = 5^2\)

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

### Step 2: Apply the power of a power rule
Using the power of a power rule \((a^m)^n = a^{m \cdot n}\):
\[ (5^6)^4 = 5^{6 \cdot 4} = 5^{24} \]
\[ (5^2)^6 = 5^{2 \cdot 6} = 5^{12} \]

Now, the expression becomes:
\[ 5^{24} \times 5^{12} \div 5^2 \]

### Step 3: Combine the products
When multiplying powers with the same base, we add the exponents:
\[ 5^{24} \times 5^{12} = 5^{24 + 12} = 5^{36} \]

### Step 4: Apply the division of powers
Now, we divide by \(5^2\):
\[ 5^{36} \div 5^2 = 5^{36 - 2} = 5^{34} \]

### Step 5: Rewrite \(25\) as a power of \(5\)
Since \(25 = 5^2\), we can rewrite the right side:
\[ (25)^{?} = (5^2)^{?} = 5^{2?} \]

### Step 6: Set the exponents equal
Now we have:
\[ 5^{34} = 5^{2?} \]
Setting the exponents equal gives:
\[ 34 = 2? \]

### Step 7: Solve for \(?\)
To find \(?\), divide both sides by 2:
\[ ? = \frac{34}{2} = 17 \]

Thus, the final answer is:
\[ ? = 17 \]

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