Solved on Jan 31, 2024

Find the value of $2^3 = 8$ .

STEP 1

Assumptions
1. We need to evaluate the expression $\frac{2}{2^{-3}}$ .

STEP 2

Understand the properties of exponents. Specifically, we will use the property that $a^{-n} = \frac{1}{a^n}$ , where $a$ is a nonzero real number and $n$ is a positive integer.

STEP 3

Apply the property of exponents to rewrite the denominator of the given expression.
$2^{-3} = \frac{1}{2^3}$

STEP 4

Now, rewrite the original expression using the result from STEP_3.
$\frac{2}{2^{-3}} = \frac{2}{\frac{1}{2^3}}$

STEP 5

Recognize that dividing by a fraction is the same as multiplying by its reciprocal.
$\frac{2}{\frac{1}{2^3}} = 2 \times 2^3$

STEP 6

Now, apply the property of exponents that states $a^m \times a^n = a^{m+n}$ , where $a$ is a nonzero real number and $m$ and $n$ are integers.

STEP 7

Since the base is the same (2), add the exponents.
$2 \times 2^3 = 2^1 \times 2^3 = 2^{1+3}$

STEP 8

Simplify the exponent.
$2^{1+3} = 2^4$

STEP 9

Calculate the value of $2^4$ .
$2^4 = 2 \times 2 \times 2 \times 2 = 16$
The value of $\frac{2}{2^{-3}}$ is 16.

Was this helpful?

Find the value of $2^3 = 8$ .

STEP 1

Assumptions
1. We need to evaluate the expression $\frac{2}{2^{-3}}$ .

STEP 2

Understand the properties of exponents. Specifically, we will use the property that $a^{-n} = \frac{1}{a^n}$ , where $a$ is a nonzero real number and $n$ is a positive integer.

STEP 3

Apply the property of exponents to rewrite the denominator of the given expression.
$2^{-3} = \frac{1}{2^3}$

STEP 4

Now, rewrite the original expression using the result from STEP_3.
$\frac{2}{2^{-3}} = \frac{2}{\frac{1}{2^3}}$

STEP 5

Recognize that dividing by a fraction is the same as multiplying by its reciprocal.
$\frac{2}{\frac{1}{2^3}} = 2 \times 2^3$

STEP 6

Now, apply the property of exponents that states $a^m \times a^n = a^{m+n}$ , where $a$ is a nonzero real number and $m$ and $n$ are integers.

STEP 7

Since the base is the same (2), add the exponents.
$2 \times 2^3 = 2^1 \times 2^3 = 2^{1+3}$

STEP 8

Simplify the exponent.
$2^{1+3} = 2^4$

STEP 9

Calculate the value of $2^4$ .
$2^4 = 2 \times 2 \times 2 \times 2 = 16$
The value of $\frac{2}{2^{-3}}$ is 16.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the value of 23=82^3 = 823=8.

STEP 1

Assumptions1. We need to evaluate the expression 22−3\frac{2}{2^{-3}}2−32​.

STEP 2

Understand the properties of exponents. Specifically, we will use the property that a−n=1ana^{-n} = \frac{1}{a^n}a−n=an1​, where aaa is a nonzero real number and nnn is a positive integer.

STEP 3

Apply the property of exponents to rewrite the denominator of the given expression.2−3=1232^{-3} = \frac{1}{2^3}2−3=231​

STEP 4

Now, rewrite the original expression using the result from STEP_3.22−3=2123\frac{2}{2^{-3}} = \frac{2}{\frac{1}{2^3}}2−32​=231​2​

STEP 5

Recognize that dividing by a fraction is the same as multiplying by its reciprocal.2123=2×23\frac{2}{\frac{1}{2^3}} = 2 \times 2^3231​2​=2×23

STEP 6

Now, apply the property of exponents that states am×an=am+na^m \times a^n = a^{m+n}am×an=am+n, where aaa is a nonzero real number and mmm and nnn are integers.

STEP 7

Since the base is the same (2), add the exponents.2×23=21×23=21+32 \times 2^3 = 2^1 \times 2^3 = 2^{1+3}2×23=21×23=21+3

STEP 8

Simplify the exponent.21+3=242^{1+3} = 2^421+3=24

STEP 9

Calculate the value of 242^424.24=2×2×2×2=162^4 = 2 \times 2 \times 2 \times 2 = 1624=2×2×2×2=16The value of 22−3\frac{2}{2^{-3}}2−32​ is 16.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the value of $2^3 = 8$ .

Assumptions
1. We need to evaluate the expression $\frac{2}{2^{-3}}$ .

Understand the properties of exponents. Specifically, we will use the property that $a^{-n} = \frac{1}{a^n}$ , where $a$ is a nonzero real number and $n$ is a positive integer.

Apply the property of exponents to rewrite the denominator of the given expression.
$2^{-3} = \frac{1}{2^3}$

Now, rewrite the original expression using the result from STEP_3.
$\frac{2}{2^{-3}} = \frac{2}{\frac{1}{2^3}}$

Recognize that dividing by a fraction is the same as multiplying by its reciprocal.
$\frac{2}{\frac{1}{2^3}} = 2 \times 2^3$

Now, apply the property of exponents that states $a^m \times a^n = a^{m+n}$ , where $a$ is a nonzero real number and $m$ and $n$ are integers.

Since the base is the same (2), add the exponents.
$2 \times 2^3 = 2^1 \times 2^3 = 2^{1+3}$

Simplify the exponent.
$2^{1+3} = 2^4$

Calculate the value of $2^4$ .
$2^4 = 2 \times 2 \times 2 \times 2 = 16$
The value of $\frac{2}{2^{-3}}$ is 16.