`4^(-3//2) + 8^(2//3)` is equal to

Options

A. `2(1)/(4)`
B. `4(1)/(8)`
C. `4(1)/(4)`
D. `8(1)/(4)`

Answer

To solve the expression \( 4^{-\frac{3}{2}} + 8^{\frac{2}{3}} \), we will follow these steps:

### Step 1: Simplify \( 4^{-\frac{3}{2}} \)
Using the property of exponents that states \( a^{-b} = \frac{1}{a^b} \), we can rewrite \( 4^{-\frac{3}{2}} \) as:
\[ 4^{-\frac{3}{2}} = \frac{1}{4^{\frac{3}{2}}} \]

### Step 2: Calculate \( 4^{\frac{3}{2}} \)
Next, we simplify \( 4^{\frac{3}{2}} \). We know that \( 4 = 2^2 \), so:
\[ 4^{\frac{3}{2}} = (2^2)^{\frac{3}{2}} = 2^{2 \cdot \frac{3}{2}} = 2^3 = 8 \]
Thus,
\[ 4^{-\frac{3}{2}} = \frac{1}{8} \]

### Step 3: Simplify \( 8^{\frac{2}{3}} \)
Now we simplify \( 8^{\frac{2}{3}} \). We know that \( 8 = 2^3 \), so:
\[ 8^{\frac{2}{3}} = (2^3)^{\frac{2}{3}} = 2^{3 \cdot \frac{2}{3}} = 2^2 = 4 \]

### Step 4: Combine the results
Now we can combine the results from Steps 2 and 3:
\[ 4^{-\frac{3}{2}} + 8^{\frac{2}{3}} = \frac{1}{8} + 4 \]

### Step 5: Convert \( 4 \) to a fraction with a common denominator
To add \( \frac{1}{8} \) and \( 4 \), we convert \( 4 \) to a fraction with a denominator of \( 8 \):
\[ 4 = \frac{32}{8} \]

### Step 6: Add the fractions
Now we can add the two fractions:
\[ \frac{1}{8} + \frac{32}{8} = \frac{1 + 32}{8} = \frac{33}{8} \]

### Final Answer
Thus, the final answer is:
\[ \frac{33}{8} \]

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