`(64)^ (-2/3) xx (1/4)^(-2)` is equal to :

Options

A. 1
B. 2
C. `1/2`
D. `1/16`

Answer

To solve the expression \((64)^{-2/3} \times (1/4)^{-2}\), we can follow these steps:

### Step 1: Rewrite the bases
First, we need to express \(64\) and \(\frac{1}{4}\) in terms of powers of \(4\).

- \(64\) can be expressed as \(4^3\) because \(4 \times 4 \times 4 = 64\).
- \(\frac{1}{4}\) can be expressed as \(4^{-1}\).

So, we rewrite the expression:
\[ (64)^{-2/3} = (4^3)^{-2/3} \]
\[ (1/4)^{-2} = (4^{-1})^{-2} \]

### Step 2: Apply the power of a power rule
Using the power of a power rule \((a^m)^n = a^{m \cdot n}\), we can simplify both parts of the expression.

- For \((4^3)^{-2/3}\):
\[ (4^3)^{-2/3} = 4^{3 \cdot (-2/3)} = 4^{-2} \]

- For \((4^{-1})^{-2}\):
\[ (4^{-1})^{-2} = 4^{-1 \cdot (-2)} = 4^{2} \]

### Step 3: Combine the results
Now, we can combine the results:
\[ 4^{-2} \times 4^{2} \]

### Step 4: Use the product of powers rule
Using the product of powers rule \(a^m \times a^n = a^{m+n}\):
\[ 4^{-2 + 2} = 4^{0} \]

### Step 5: Evaluate the expression
We know that any non-zero number raised to the power of \(0\) is \(1\):
\[ 4^{0} = 1 \]

Thus, the final answer is:
\[ \boxed{1} \]

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