`(562.5xx6)^(6)div(135div9)^(10)div (37.5xx6)^(7)=(3.75xx4)^(?-6)`

Options

A. 0
B. 2
C. 3
D. 4

Explanation

To solve the equation

\[ (562.5 \times 6)^{6} \div (135 \div 9)^{10} \div (37.5 \times 6)^{7} = (3.75 \times 4)^{? - 6} \]

we will follow these steps:

### Step 1: Simplify the left-hand side

First, we simplify \( 135 \div 9 \):

\[ 135 \div 9 = 15 \]

So the equation becomes:

\[ (562.5 \times 6)^{6} \div 15^{10} \div (37.5 \times 6)^{7} \]

### Step 2: Calculate \( 562.5 \times 6 \)

Next, we calculate \( 562.5 \times 6 \):

\[ 562.5 \times 6 = 3375 \]

Now, we can rewrite the equation:

\[ (3375)^{6} \div 15^{10} \div (37.5 \times 6)^{7} \]

### Step 3: Calculate \( 37.5 \times 6 \)

Now, we calculate \( 37.5 \times 6 \):

\[ 37.5 \times 6 = 225 \]

So the equation now looks like:

\[ (3375)^{6} \div 15^{10} \div (225)^{7} \]

### Step 4: Rewrite \( 3375 \) and \( 225 \)

Next, we can express \( 3375 \) and \( 225 \) in terms of \( 15 \):

\[ 3375 = 15^{3} \quad \text{(since } 15^3 = 3375\text{)} \]
\[ 225 = 15^{2} \quad \text{(since } 15^2 = 225\text{)} \]

Now substituting these values back into the equation:

\[ (15^{3})^{6} \div 15^{10} \div (15^{2})^{7} \]

### Step 5: Simplify the powers

Now we simplify the powers:

\[ 15^{18} \div 15^{10} \div 15^{14} \]

This can be simplified using the property of exponents:

\[ 15^{18 - 10 - 14} = 15^{18 - 24} = 15^{-6} \]

### Step 6: Set the equation equal to the right-hand side

Now we have:

\[ 15^{-6} = (3.75 \times 4)^{? - 6} \]

### Step 7: Calculate \( 3.75 \times 4 \)

Now we calculate \( 3.75 \times 4 \):

\[ 3.75 \times 4 = 15 \]

So we rewrite the equation:

\[ 15^{-6} = 15^{? - 6} \]

### Step 8: Set the exponents equal to each other

Since the bases are the same, we can set the exponents equal to each other:

\[ -6 = ? - 6 \]

### Step 9: Solve for \( ? \)

Now, we solve for \( ? \):

\[ ? = -6 + 6 = 0 \]

### Final Answer

Thus, the value of \( ? \) is:

\[ \boxed{0} \]

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