Exponent Properties

Table of Exponent Values for Base Two

$2^n$	Exponentiation Result
$2^1$	$2$
$2^2$	$4$
$2^3$	$8$
$2^4$	$16$
$2^5$	$32$
$2^6$	$64$
$2^7$	$128$
$2^8$	$256$
$2^9$	$512$
$2^{10}$	$1024$

Exponent Properties

There are several exponent properties that need to be understood:

$a^m \cdot a^n = a^{m+n}$ , where $a \neq 0, m, n$ are integers

This means multiplying two exponents with the same base results in a new exponent with the sum of the powers.
$\frac{a^m}{a^n} = a^{m-n}$ , where $a \neq 0, m, n$ are integers

Dividing two exponents with the same base results in a new exponent with the difference of the powers.
$(a^m)^n = a^{m \times n}$ , where $a \neq 0, m, n$ are integers

An exponent of an exponent means multiplying the power by the outer power.
$(ab)^m = a^m \times b^m$ , where $a, b \neq 0$ , and $m$ is an integer

The exponent of a multiplication equals the multiplication of each base raised to the same power.
$\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$ , where $b \neq 0$ , and $m$ is an integer

The exponent of a division equals the division of each base raised to the same power.
$(a^\frac{m}{n})^p \cdot (a^\frac{p}{n}) = (a)^\frac{m+p}{n}$
, where $a > 0$ , $\frac{m}{n}$ and $\frac{p}{n}$ are rational numbers with $n \neq 0$
$(a^\frac{m}{n}) \cdot (a^\frac{p}{q}) = (a)^\frac{m \cdot q + p \cdot n}{n \cdot q}$
, where $a > 0$ , $\frac{m}{n}$ and $\frac{p}{q}$ are rational numbers with $n, q \neq 0$

Importance of Conditions for Each Property

Each exponent property has specific conditions:

In properties $1$ , $2$ , and $3$ , the value $a \neq 0$ because exponents with base $0$ are only defined for positive powers.
In property $4$ , the values $a, b \neq 0$ to ensure the exponent is defined.
In property $5$ , the value $b \neq 0$ to avoid division by zero.
In properties $6$ and $7$ , the value $a > 0$ because rational exponents on negative numbers can result in complex numbers.

Understanding these exponent properties is very important as a foundation for advanced mathematics learning, such as logarithms, exponential functions, and calculus.

Worked Examples

Simplify $\frac{2^5 \times 2^3}{2^2}$

$\frac{2^5 \times 2^3}{2^2} = \frac{2^{5+3}}{2^2} = \frac{2^8}{2^2} = 2^{8-2} = 2^6 = 64$
Simplify $2^2 \cdot 2^3$

$2^2 \cdot 2^3 = 2^{2+3} = 2^5 = 32$
Simplify $2^5 \cdot 2^2$

$2^5 \cdot 2^2 = 2^{5+2} = 2^7 = 128$
Simplify $2^3 \cdot 2^7$

$2^3 \cdot 2^7 = 2^{3+7} = 2^{10} = 1024$
Simplify $\frac{2^8}{2^6}$

$\frac{2^8}{2^6} = 2^{8-6} = 2^2 = 4$
Simplify $\frac{2^{10}}{2^3}$

$\frac{2^{10}}{2^3} = 2^{10-3} = 2^7 = 128$
Simplify $\frac{2^6}{2^4}$

$\frac{2^6}{2^4} = 2^{6-4} = 2^2 = 4$
Simplify $(2^3)^3$

$(2^3)^3 = 2^{3 \times 3} = 2^9 = 512$
Simplify $(2^4)^2$

$(2^4)^2 = 2^{4 \times 2} = 2^8 = 256$
Simplify $(2^2)^5$

$(2^2)^5 = 2^{2 \times 5} = 2^{10} = 1024$

2^n

Exponentiation Result

2^1

2

2^2

4

2^3

8

2^4

16

2^5

32

2^6

64

2^7

128

2^8

256

2^9

512

2^{10}

1024

Exponent Properties

There are several exponent properties that need to be understood:

$a^m \cdot a^n = a^{m+n}$ , where $a \neq 0, m, n$ are integers

This means multiplying two exponents with the same base results in a new exponent with the sum of the powers.

$\frac{a^m}{a^n} = a^{m-n}$ , where $a \neq 0, m, n$ are integers

Dividing two exponents with the same base results in a new exponent with the difference of the powers.

$(a^m)^n = a^{m \times n}$ , where $a \neq 0, m, n$ are integers

An exponent of an exponent means multiplying the power by the outer power.

$(ab)^m = a^m \times b^m$ , where $a, b \neq 0$ , and $m$ is an integer

The exponent of a multiplication equals the multiplication of each base raised to the same power.

$\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$ , where $b \neq 0$ , and $m$ is an integer

The exponent of a division equals the division of each base raised to the same power.

(a^\frac{m}{n})^p \cdot (a^\frac{p}{n}) = (a)^\frac{m+p}{n}

, where $a > 0$ , $\frac{m}{n}$ and $\frac{p}{n}$ are rational numbers with $n \neq 0$

(a^\frac{m}{n}) \cdot (a^\frac{p}{q}) = (a)^\frac{m \cdot q + p \cdot n}{n \cdot q}

, where $a > 0$ , $\frac{m}{n}$ and $\frac{p}{q}$ are rational numbers with $n, q \neq 0$

Importance of Conditions for Each Property

Each exponent property has specific conditions:

In properties

1

2

, and

3

, the value

a \neq 0

because exponents with base

0

are only defined for positive powers.

In property

4

, the values

a, b \neq 0

to ensure the exponent is defined.

In property

5

, the value

b \neq 0

to avoid division by zero.

In properties

6

and

7

, the value

a > 0

because rational exponents on negative numbers can result in complex numbers.

Understanding these exponent properties is very important as a foundation for advanced mathematics learning, such as logarithms, exponential functions, and calculus.

Worked Examples

Simplify $\frac{2^5 \times 2^3}{2^2}$

\frac{2^5 \times 2^3}{2^2} = \frac{2^{5+3}}{2^2} = \frac{2^8}{2^2} = 2^{8-2} = 2^6 = 64

Simplify $2^2 \cdot 2^3$

2^2 \cdot 2^3 = 2^{2+3} = 2^5 = 32

Simplify $2^5 \cdot 2^2$

2^5 \cdot 2^2 = 2^{5+2} = 2^7 = 128

Simplify $2^3 \cdot 2^7$

2^3 \cdot 2^7 = 2^{3+7} = 2^{10} = 1024

Simplify $\frac{2^8}{2^6}$

\frac{2^8}{2^6} = 2^{8-6} = 2^2 = 4

Simplify $\frac{2^{10}}{2^3}$

\frac{2^{10}}{2^3} = 2^{10-3} = 2^7 = 128

Simplify $\frac{2^6}{2^4}$

\frac{2^6}{2^4} = 2^{6-4} = 2^2 = 4

Simplify $(2^3)^3$

(2^3)^3 = 2^{3 \times 3} = 2^9 = 512

Simplify $(2^4)^2$

(2^4)^2 = 2^{4 \times 2} = 2^8 = 256

Simplify $(2^2)^5$

(2^2)^5 = 2^{2 \times 5} = 2^{10} = 1024