Find the value of n from the given equation. `5^(12) div root(4) (625) = 5^(3n-1)`

Options

A. `4^(-1)`
B. `4^2`
C. `4^(-2)`
D. 4

Answer

To solve the equation \( 5^{12} \div \sqrt[4]{625} = 5^{3n-1} \), we will follow these steps:

### Step 1: Simplify the left side of the equation
We need to find the value of \( \sqrt[4]{625} \).

### Step 2: Express 625 as a power of 5
We know that:
\[ 625 = 5^4 \]

### Step 3: Calculate the fourth root of 625
Using the property of exponents:
\[ \sqrt[4]{625} = \sqrt[4]{5^4} = 5^{4 \cdot \frac{1}{4}} = 5^1 = 5 \]

### Step 4: Substitute back into the equation
Now substitute \( \sqrt[4]{625} \) back into the equation:
\[ 5^{12} \div 5 = 5^{12 - 1} = 5^{11} \]

### Step 5: Set the equation equal to the right side
Now we have:
\[ 5^{11} = 5^{3n - 1} \]

### Step 6: Since the bases are the same, equate the exponents
This gives us:
\[ 11 = 3n - 1 \]

### Step 7: Solve for \( n \)
Add 1 to both sides:
\[ 11 + 1 = 3n \]
\[ 12 = 3n \]

Now, divide both sides by 3:
\[ n = \frac{12}{3} = 4 \]

### Final Answer
The value of \( n \) is \( 4 \).

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