If `7^(m+1)` = 2401, then find the value of `2^(2m+2)`

Options

A. 224
B. 256
C. 264
D. 286

Answer

To solve the equation \( 7^{m+1} = 2401 \) and find the value of \( 2^{2m+2} \), we can follow these steps:

### Step 1: Rewrite 2401 as a power of 7
We know that \( 2401 \) can be expressed as a power of \( 7 \). Let's find out which power of \( 7 \) it is.

\[ 2401 = 7^4 \]

### Step 2: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:

\[ m + 1 = 4 \]

### Step 3: Solve for \( m \)
Now, we can solve for \( m \):

\[ m = 4 - 1 = 3 \]

### Step 4: Substitute \( m \) into \( 2^{2m+2} \)
Now that we have \( m \), we can substitute it into the expression \( 2^{2m+2} \):

\[ 2^{2m+2} = 2^{2(3)+2} \]

### Step 5: Simplify the exponent
Now simplify the exponent:

\[ 2^{2(3)+2} = 2^{6+2} = 2^8 \]

### Step 6: Calculate \( 2^8 \)
Finally, we calculate \( 2^8 \):

\[ 2^8 = 256 \]

### Final Answer
Thus, the value of \( 2^{2m+2} \) is \( 256 \).

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