simplify `64^((2)/(3)) xx 64^((1)/(3)) xx 64^((-5)/(3))=` _____

Options

A. `(1)/(64)`
B. `(1)/(16)`
C. 32
D. `(1)/(32)`

Answer

To simplify the expression \( 64^{\frac{2}{3}} \times 64^{\frac{1}{3}} \times 64^{-\frac{5}{3}} \), we can follow these steps:

### Step 1: Apply the property of indices
We know that when multiplying powers with the same base, we can add the exponents. Therefore, we can rewrite the expression as:
\[ 64^{\frac{2}{3} + \frac{1}{3} - \frac{5}{3}} \]

### Step 2: Simplify the exponent
Now, we need to simplify the exponent:
\[ \frac{2}{3} + \frac{1}{3} - \frac{5}{3} = \frac{2 + 1 - 5}{3} = \frac{-2}{3} \]

### Step 3: Rewrite the expression
Now, substituting the simplified exponent back into the expression, we have:
\[ 64^{-\frac{2}{3}} \]

### Step 4: Convert to a positive exponent
Using the property that \( a^{-n} = \frac{1}{a^n} \), we can rewrite the expression as:
\[ \frac{1}{64^{\frac{2}{3}}} \]

### Step 5: Simplify \( 64^{\frac{2}{3}} \)
Next, we need to simplify \( 64^{\frac{2}{3}} \). We can express \( 64 \) as \( 4^3 \):
\[ 64^{\frac{2}{3}} = (4^3)^{\frac{2}{3}} = 4^{3 \cdot \frac{2}{3}} = 4^2 \]

### Step 6: Calculate \( 4^2 \)
Now, we calculate \( 4^2 \):
\[ 4^2 = 16 \]

### Step 7: Final expression
Thus, we have:
\[ 64^{-\frac{2}{3}} = \frac{1}{16} \]

### Final Answer
The simplified expression is:
\[ \frac{1}{16} \]

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