What will come in place of the question mark (?) in the given questions ?
`(125^2xxsqrt((15))xx5^3)/sqrt((375))=5^(11-?)`

Options

A. 6
B. 8
C. 3
D. 4

Answer

To solve the equation

\[ \frac{(125^2 \cdot \sqrt{15} \cdot 5^3)}{\sqrt{375}} = 5^{11 - ?} \]

we will simplify the left-hand side step by step.

### Step 1: Rewrite 125 and 375 in terms of powers of 5
We know that:
- \(125 = 5^3\)
- \(375 = 25 \cdot 15 = 5^2 \cdot 15\)

### Step 2: Substitute these values into the equation
Now substituting these values into the equation gives us:

\[ \frac{(5^3)^2 \cdot \sqrt{15} \cdot 5^3}{\sqrt{5^2 \cdot 15}} \]

### Step 3: Simplify the powers
Calculating the powers, we have:

\[ (5^3)^2 = 5^{3 \cdot 2} = 5^6 \]

So the equation becomes:

\[ \frac{5^6 \cdot \sqrt{15} \cdot 5^3}{\sqrt{5^2 \cdot 15}} \]

### Step 4: Combine the powers of 5 in the numerator
Now, we can combine the powers of 5 in the numerator:

\[ 5^6 \cdot 5^3 = 5^{6 + 3} = 5^9 \]

Thus, the equation is now:

\[ \frac{5^9 \cdot \sqrt{15}}{\sqrt{5^2 \cdot 15}} \]

### Step 5: Simplify the denominator
Now, simplify the denominator:

\[ \sqrt{5^2 \cdot 15} = \sqrt{5^2} \cdot \sqrt{15} = 5 \cdot \sqrt{15} \]

### Step 6: Substitute back into the equation
Now substituting this back into the equation gives us:

\[ \frac{5^9 \cdot \sqrt{15}}{5 \cdot \sqrt{15}} = \frac{5^9}{5} = 5^{9 - 1} = 5^8 \]

### Step 7: Set the left-hand side equal to the right-hand side
Now we have:

\[ 5^8 = 5^{11 - ?} \]

### Step 8: Equate the exponents
Since the bases are the same, we can equate the exponents:

\[ 8 = 11 - ? \]

### Step 9: Solve for ?
Rearranging gives us:

\[ ? = 11 - 8 = 3 \]

Thus, the value that comes in place of the question mark is:

\[ \boxed{3} \]

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