Solve the following expressions .
`(2^2 + 2^3 + 2^(-2) + 2^(-3))`

Answer

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

### Step 1: Evaluate the positive powers of 2
Calculate \(2^2\) and \(2^3\):
\[ 2^2 = 4 \]
\[ 2^3 = 8 \]

### Step 2: Convert the negative powers to positive
For the negative powers, we use the property that \(a^{-n} = \frac{1}{a^n}\):
\[ 2^{-2} = \frac{1}{2^2} = \frac{1}{4} \]
\[ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \]

### Step 3: Rewrite the expression
Now we can rewrite the original expression with the calculated values:
\[ 4 + 8 + \frac{1}{4} + \frac{1}{8} \]

### Step 4: Combine the whole numbers
First, add the whole numbers:
\[ 4 + 8 = 12 \]

### Step 5: Find a common denominator for the fractions
The fractions \(\frac{1}{4}\) and \(\frac{1}{8}\) need a common denominator to be added. The least common multiple (LCM) of 4 and 8 is 8. Rewrite \(\frac{1}{4}\) with a denominator of 8:
\[ \frac{1}{4} = \frac{2}{8} \]
Now we can add the fractions:
\[ \frac{2}{8} + \frac{1}{8} = \frac{3}{8} \]

### Step 6: Combine the whole number with the fraction
Now, combine the whole number with the fraction:
\[ 12 + \frac{3}{8} = 12 \frac{3}{8} \]

### Final Answer
Thus, the final answer to the expression \(2^2 + 2^3 + 2^{-2} + 2^{-3}\) is:
\[ 12 \frac{3}{8} \]

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