What is the value of x, if `64xx(512)^(2)=x^8`?

Options

A. `5`
B. `7`
C. `2`
D. `8`

Answer

To solve the equation \( 64 \times (512)^2 = x^8 \), we will follow these steps:

### Step 1: Rewrite the numbers in terms of powers of 8
First, we need to express \( 64 \) and \( 512 \) as powers of \( 8 \).

- \( 64 = 8^2 \) (since \( 8^2 = 64 \))
- \( 512 = 8^3 \) (since \( 8^3 = 512 \))

### Step 2: Substitute the powers into the equation
Now, we can substitute these values back into the equation:

\[ 64 \times (512)^2 = x^8 \]

becomes

\[ 8^2 \times (8^3)^2 = x^8 \]

### Step 3: Simplify the equation
Next, simplify \( (8^3)^2 \):

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

Now substitute this back into the equation:

\[ 8^2 \times 8^6 = x^8 \]

### Step 4: Combine the powers of 8
Using the property of exponents that states \( a^m \times a^n = a^{m+n} \):

\[ 8^2 \times 8^6 = 8^{2+6} = 8^8 \]

So, we have:

\[ x^8 = 8^8 \]

### Step 5: Solve for x
To find \( x \), we take the eighth root of both sides:

\[ x = 8 \]

### Final Answer
Thus, the value of \( x \) is \( 8 \).

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