Find the value of `(4)/((216)^(-2//3))-(1)/((256)^(-3//4))`

Options

A. `144`
B. `64`
C. `80`
D. `36`

Answer

To solve the expression \( \frac{4}{(216)^{-\frac{2}{3}}} - \frac{1}{(256)^{-\frac{3}{4}}} \), we will follow these steps:

### Step 1: Simplify the negative exponents
First, we rewrite the terms with negative exponents as follows:
\[ \frac{4}{(216)^{-\frac{2}{3}}} = 4 \cdot (216)^{\frac{2}{3}} \]
\[ \frac{1}{(256)^{-\frac{3}{4}}} = (256)^{\frac{3}{4}} \]
So, we can rewrite the expression as:
\[ 4 \cdot (216)^{\frac{2}{3}} - (256)^{\frac{3}{4}} \]

### Step 2: Prime factorization
Next, we need to find the prime factorization of 216 and 256:
- \( 216 = 6^3 \) (since \( 6 \times 6 \times 6 = 216 \))
- \( 256 = 4^4 \) (since \( 4 \times 4 \times 4 \times 4 = 256 \))

### Step 3: Substitute the factorizations
Now we substitute these factorizations back into the expression:
\[ 4 \cdot (6^3)^{\frac{2}{3}} - (4^4)^{\frac{3}{4}} \]

### Step 4: Simplify the powers
Using the power of a power property \((a^m)^n = a^{m \cdot n}\):
\[ (6^3)^{\frac{2}{3}} = 6^{3 \cdot \frac{2}{3}} = 6^2 \]
\[ (4^4)^{\frac{3}{4}} = 4^{4 \cdot \frac{3}{4}} = 4^3 \]
So our expression now looks like:
\[ 4 \cdot 6^2 - 4^3 \]

### Step 5: Calculate the powers
Now we calculate \(6^2\) and \(4^3\):
\[ 6^2 = 36 \]
\[ 4^3 = 64 \]

### Step 6: Substitute and simplify
Substituting these values back into the expression gives:
\[ 4 \cdot 36 - 64 \]
Calculating \(4 \cdot 36\):
\[ 4 \cdot 36 = 144 \]
Thus, we have:
\[ 144 - 64 \]

### Step 7: Final calculation
Finally, we perform the subtraction:
\[ 144 - 64 = 80 \]

### Final Answer
The value of the expression is \(80\).

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