Find the value of m, if `((9^(0)+7^0)xx(9+7))/(5^(0)+3^(0)+9^(0)+7^(0))=(2^(5m+4))/((2^2)^(2m+1))`

Options

A. `1`
B. `4`
C. `3`
D. `-5`

Answer

To solve the equation

\[ \frac{(9^0 + 7^0) \times (9 + 7)}{(5^0 + 3^0 + 9^0 + 7^0)} = \frac{2^{5m + 4}}{(2^2)^{2m + 1}}, \]

we will follow these steps:

### Step 1: Simplify the left-hand side
First, we know that any number raised to the power of 0 is equal to 1. Therefore:

\[ 9^0 = 1 \quad \text{and} \quad 7^0 = 1. \]

So, we can substitute these values into the equation:

\[ (1 + 1) \times (9 + 7) = 2 \times 16 = 32. \]

### Step 2: Simplify the denominator of the left-hand side
Now, we simplify the denominator:

\[ 5^0 = 1, \quad 3^0 = 1, \quad 9^0 = 1, \quad 7^0 = 1. \]

Thus,

\[ (1 + 1 + 1 + 1) = 4. \]

### Step 3: Combine the left-hand side
Now, we can combine the left-hand side:

\[ \frac{32}{4} = 8. \]

### Step 4: Simplify the right-hand side
Now, let's simplify the right-hand side:

\[ (2^2)^{2m + 1} = 2^{2(2m + 1)} = 2^{4m + 2}. \]

So, the right-hand side becomes:

\[ \frac{2^{5m + 4}}{2^{4m + 2}} = 2^{(5m + 4) - (4m + 2)} = 2^{m + 2}. \]

### Step 5: Set the left-hand side equal to the right-hand side
Now we have:

\[ 8 = 2^{m + 2}. \]

### Step 6: Express 8 as a power of 2
We know that:

\[ 8 = 2^3. \]

### Step 7: Set the exponents equal to each other
Now we can set the exponents equal to each other:

\[ m + 2 = 3. \]

### Step 8: Solve for m
Subtract 2 from both sides:

\[ m = 3 - 2 = 1. \]

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

\[ \boxed{1}. \]

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