`(512)^((-2)/(3) ) xx ((1)/(4))^(-3)` is equal to

Options

A. 4
B. `1/4`
C. 1
D. 16

Answer

To solve the expression \((512)^{-\frac{2}{3}} \times \left(\frac{1}{4}\right)^{-3}\), we will break it down step by step.

### Step 1: Rewrite the bases in terms of powers of 2
First, we need to express \(512\) and \(\frac{1}{4}\) as powers of \(2\).

- \(512 = 2^9\) (since \(2^9 = 512\))
- \(\frac{1}{4} = 4^{-1} = (2^2)^{-1} = 2^{-2}\)

### Step 2: Substitute the values into the expression
Now we can rewrite the expression using these powers of \(2\):

\[ (512)^{-\frac{2}{3}} = (2^9)^{-\frac{2}{3}} \quad \text{and} \quad \left(\frac{1}{4}\right)^{-3} = (2^{-2})^{-3} \]

So the expression becomes:

\[ (2^9)^{-\frac{2}{3}} \times (2^{-2})^{-3} \]

### Step 3: Apply the power of a power property
Using the property \((a^m)^n = a^{m \cdot n}\):

\[ (2^9)^{-\frac{2}{3}} = 2^{9 \cdot -\frac{2}{3}} = 2^{-6} \]
\[ (2^{-2})^{-3} = 2^{-2 \cdot -3} = 2^6 \]

### Step 4: Combine the results
Now we can combine the two results:

\[ 2^{-6} \times 2^6 \]

Using the property \(a^m \times a^n = a^{m+n}\):

\[ 2^{-6 + 6} = 2^0 \]

### Step 5: Simplify the expression
Since \(2^0 = 1\):

\[ 2^0 = 1 \]

### Final Answer
Thus, the value of the expression \((512)^{-\frac{2}{3}} \times \left(\frac{1}{4}\right)^{-3}\) is equal to \(1\).

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