`4^(x)=root(7)(1024)` then what is the value of x?

Options

A. `5//7`
B. `4//7`
C. `3//7`
D. `6//7`

Answer

To solve the equation \( 4^x = \sqrt[7]{1024} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 4^x = \sqrt[7]{1024} \]

### Step 2: Express 4 and 1024 as powers of 2
We know that:
\[ 4 = 2^2 \quad \text{and} \quad 1024 = 2^{10} \]
Thus, we can rewrite the equation as:
\[ (2^2)^x = \sqrt[7]{2^{10}} \]

### Step 3: Simplify both sides
Using the power of a power property, we simplify the left side:
\[ 2^{2x} = \sqrt[7]{2^{10}} \]
The right side can be rewritten using the property of roots:
\[ \sqrt[7]{2^{10}} = 2^{10/7} \]
So now we have:
\[ 2^{2x} = 2^{10/7} \]

### Step 4: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal:
\[ 2x = \frac{10}{7} \]

### Step 5: Solve for x
Now, we solve for \( x \) by dividing both sides by 2:
\[ x = \frac{10}{7} \cdot \frac{1}{2} = \frac{10}{14} = \frac{5}{7} \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{\frac{5}{7}} \]