Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Jan 24, 2024

Evaluate $2^{x}$ for (a) $x=-2$ and (b) $x=3$

STEP 1

Assumptions
1. We are evaluating the expression $2^{x}$ .
2. We will evaluate it for two different values of $x$ : $x=-2$ and $x=3$ .

STEP 2

First, we will evaluate $2^{x}$ for $x=-2$ . To do this, we recall that a negative exponent indicates a reciprocal.
$2^{x} = 2^{-2}$

STEP 3

We rewrite $2^{-2}$ as the reciprocal of $2^{2}$ .
$2^{-2} = \frac{1}{2^{2}}$

STEP 4

Now we calculate $2^{2}$ .
$2^{2} = 2 \times 2 = 4$

STEP 5

Substitute the value of $2^{2}$ into the reciprocal expression.
$2^{-2} = \frac{1}{4}$

STEP 6

Thus, $2^{x}$ evaluated at $x=-2$ is $\frac{1}{4}$ .

STEP 7

Next, we will evaluate $2^{x}$ for $x=3$ . This is a straightforward calculation since the exponent is positive.
$2^{x} = 2^{3}$

STEP 8

Calculate $2^{3}$ .
$2^{3} = 2 \times 2 \times 2 = 8$

STEP 9

Thus, $2^{x}$ evaluated at $x=3$ is $8$ .
The evaluations are: (a) For $x=-2$ , $2^{x} = \frac{1}{4}$ (b) For $x=3$ , $2^{x} = 8$

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