Simplify:- (5*5*5*5*5*5)^4 * (5*5)^6 ÷ (5)^2 = (25)^?

Options

A. a. 10
B. b. 20
C. c. 17
D. d. 19

Answer

To simplify the expression \((5*5*5*5*5*5)^4 * (5*5)^6 ÷ (5)^2\) and express it in the form \((25)^?\), we can follow these steps:

### Step 1: Rewrite the expression using exponents
The expression can be rewritten using exponents:
\[ (5^6)^4 * (5^2)^6 ÷ (5^2) \]

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

### Step 3: Combine the exponents for multiplication
When multiplying with the same base, we add the exponents:
\[ 5^{24 + 12} ÷ 5^2 = 5^{36} ÷ 5^2 \]

### Step 4: Apply the quotient rule
Using the quotient rule \(\frac{a^m}{a^n} = a^{m-n}\):
\[ 5^{36 - 2} = 5^{34} \]

### Step 5: Convert to base 25
Since \(25 = 5^2\), we can express \(5^{34}\) in terms of \(25\):
\[ 5^{34} = (5^2)^{17} = 25^{17} \]

### Conclusion
Thus, we have:
\[ (5*5*5*5*5*5)^4 * (5*5)^6 ÷ (5)^2 = 25^{17} \]
So, the value of the question mark is \(17\).