What will come in place of the question mark (?) in the following questions ?
`216^(1/3)xx26^4xx39^4-:(12^4xx3xx2^-3]=13^?`

Options

A. 8
B. 12
C. 4
D. 10

Answer

To solve the equation \( 216^{1/3} \times 26^4 \times 39^4 \div (12^4 \times 3 \times 2^{-3}) = 13^{?} \), we will follow these steps:

### Step 1: Simplify \( 216^{1/3} \)
First, we recognize that \( 216 = 6^3 \). Thus, we can rewrite:
\[ 216^{1/3} = (6^3)^{1/3} = 6^{3 \times \frac{1}{3}} = 6^1 = 6 \]

**Hint:** Remember that \( a^{m/n} = (a^m)^{1/n} \).

### Step 2: Rewrite \( 26^4 \) and \( 39^4 \)
Next, we rewrite \( 26 \) and \( 39 \) in terms of their prime factors:
\[ 26 = 13 \times 2 \quad \text{and} \quad 39 = 13 \times 3 \]
Thus:
\[ 26^4 = (13 \times 2)^4 = 13^4 \times 2^4 \]
\[ 39^4 = (13 \times 3)^4 = 13^4 \times 3^4 \]

### Step 3: Combine the terms
Now, substituting back into the equation:
\[ 6 \times (13^4 \times 2^4) \times (13^4 \times 3^4) = 6 \times 13^8 \times 2^4 \times 3^4 \]

**Hint:** Use the property of exponents to combine like bases.

### Step 4: Simplify the denominator
Next, we simplify the denominator \( 12^4 \times 3 \times 2^{-3} \):
\[ 12 = 3 \times 4 = 3 \times 2^2 \quad \Rightarrow \quad 12^4 = (3 \times 2^2)^4 = 3^4 \times 2^8 \]
Thus:
\[ 12^4 \times 3 \times 2^{-3} = (3^4 \times 2^8) \times 3 \times 2^{-3} = 3^{4+1} \times 2^{8-3} = 3^5 \times 2^5 \]

### Step 5: Write the full equation
Now we can write the full equation:
\[ \frac{6 \times 13^8 \times 2^4 \times 3^4}{3^5 \times 2^5} = 13^{?} \]

### Step 6: Simplify the left side
Now, we simplify the left side:
\[ 6 = 2 \times 3 \quad \Rightarrow \quad \frac{(2 \times 3) \times 13^8 \times 2^4 \times 3^4}{3^5 \times 2^5} = \frac{2^{1+4} \times 3^{1+4} \times 13^8}{3^5 \times 2^5} = \frac{2^5 \times 3^5 \times 13^8}{3^5 \times 2^5} \]
The \( 2^5 \) and \( 3^5 \) cancel out:
\[ = 13^8 \]

### Step 7: Set the equation
Now we have:
\[ 13^8 = 13^{?} \]
Thus, we can conclude that:
\[ ? = 8 \]

### Final Answer
The value that comes in place of the question mark (?) is \( 8 \).

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