If `5 ^(2x +1) div 25 = 125,` find the value of `x`.

Options

A. 6
B. 4
C. 2
D. `-2`

Answer

To solve the equation \( \frac{5^{(2x + 1)}}{25} = 125 \), we will follow these steps:

### Step 1: Rewrite the equation
We know that \( 25 \) can be expressed as \( 5^2 \) and \( 125 \) can be expressed as \( 5^3 \). Therefore, we can rewrite the equation as:
\[ \frac{5^{(2x + 1)}}{5^2} = 5^3 \]

### Step 2: Apply the property of exponents
Using the property of exponents that states \( \frac{a^m}{a^n} = a^{m-n} \), we can simplify the left side:
\[ 5^{(2x + 1 - 2)} = 5^3 \]
This simplifies to:
\[ 5^{(2x - 1)} = 5^3 \]

### Step 3: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ 2x - 1 = 3 \]

### Step 4: Solve for \( x \)
Now, we will solve for \( x \). First, add \( 1 \) to both sides:
\[ 2x = 3 + 1 \]
\[ 2x = 4 \]
Next, divide both sides by \( 2 \):
\[ x = \frac{4}{2} \]
\[ x = 2 \]

### Final Answer
The value of \( x \) is \( 2 \).

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