Exponent

Summary: Exponent Rules

a^m⋅aⁿ = a^{m + n}
Example
(a^m)ⁿ = a^m⋅n
Example
(ab)^m = a^m⋅b^m
Example
a^m
aⁿ
= a^{m - n}
Example
(

a
b

)
^m
=

a^m
b^m

Example
a⁰ = 1
Example
a^-m =

1
a^m

Example

Product of Powers

Formula

a^m⋅aⁿ = a^{m + n}

Example

Simplify the given expression.

x⁶⋅x⁸

Solution

1

x⁶⋅x⁸ = x^{6 + 8}
2

= x¹⁴

x⁶ and x⁸ have the same base: x.
So use the product of powers rule.

Example

Simplify the given expression.

3²xy⁴⋅3x³y⁷

Solution

1

3² x y⁴⋅3 x³ y⁷
2

= 3^{2 + 1}⋅x^{1 + 3}⋅y^{4 + 7}
3

= 3³⋅x⁴⋅y¹¹
4

= 27x⁴y¹¹

1 ~ 2:

Multiply the same base numbers.

Power of a Power

Formula

(a^m)ⁿ = a^m⋅n

Example

Simplify the given expression.

(x²)³

Solution

1

(x²)³ = x^2⋅3
2

= x⁶

Example

Simplify the given expression.

((x⁴)²)⁵

Solution

1

((x⁴)²)⁵
2

= (x^4⋅2)⁵
3

= (x⁸)⁵
4

= x^8⋅5
5

= x⁴⁰

1 ~ 2:

Solve the inner power.

3 ~ 4:

Then solve the outer power.

Power of a Product

Formula

(ab)^m = a^m⋅b^m

Example

Simplify the given expression.

(x³y)²

Solution

1

(x³y)²
2

= (x³)²⋅y²
3

= x⁶y²

2 ~ 3:

(x³)²
= x^3⋅2
= x⁶

Power of a Power

Quotient of Powers

Formula

a^m

aⁿ

= a^{m - n}

Example

Simplify the given expression. (x ≠ 0)

x⁸

x³

Solution

1

x⁸
x³

= x^{8 - 3}
2

= x⁵

x⁸ and x³ have the same base: x.
So use the quotient of powers rule.

Example

Simplify the given expression. (x, y, z ≠ 0)

12x⁷y²z⁹

4x⁴y²z⁵

Solution

1

12 x⁷ y² z⁹
4 x⁴ y² z⁵
2

= 3⋅x^{7 - 4}⋅z^{9 - 5}
3

= 3x³z⁴

1 ~ 2:

12/3 = 4
And cancel y².

Power of a Quotient

Formula

(

)

a^m

b^m

Example

Simplify the given expression. (y ≠ 0)

(

y²

)

Solution

1

(

x
y²

)
³
2

=

x²
(y²)³
3

=

x²
y⁶

2 ~ 3:

(y²)³
= y^2⋅3
= y⁶

Power of a Power

Example

Simplify the given expression. (x, y ≠ 0)

(

3x⁴

)

Solution

1

(

3x⁴
y

)
²
⋅

(

y
2x

)
³
2

=

3²⋅(x⁴)²
y²

⋅

y³
2³⋅x³
3

=

9⋅x⁸
y²

⋅

y³
8⋅x³
4

=

9x⁵y
8

2 ~ 3:

3² = 9
2³ = 8

(x⁴)²
= x^4⋅2
= x⁸

Power of a Power

3 ~ 4:

x⁸/x³
= x^{8 - 3}
= x⁵

y³/y²
= y^{3 - 2}
= y¹
= y

Quotient of Powers

Zero Exponent

Formula

a⁰ = 1

Example

Simplify the given expression.

2⁰

Solution

1

2⁰ = 1

Example

Simplify the given expression. (x, y ≠ 0)

4x²y⁰⋅(-3xy)⁰

Solution

1

4x² y⁰⋅(-3xy)⁰
2

= 4x²⋅1⋅1
3

= 4x²

Negative Exponent

Formula

a^-m =

a^m

Example

Simplify the given expression. (x ≠ 0)

x^-4

Solution

1

x^-4 =

1
x⁴

If the exponent is (-),
switch the number's position,
(numerator) ↔ (denominator),
and change the sign of the exponent to (+).

x^-4
The exponent is (-).
So move x^-4
from the numerator to the denominator,
and change the exponent -4 to 4.

Example

Simplify the given expression. (x ≠ 0)

x^-3

Solution

1

1
x^-3

=

x³
1
2

= x³

x^-3
The exponent is (-).
So move x^-3
from the denominator to the numerator,
and change the exponent -3 to 3.

Example

Simplify the given expression. (x, y ≠ 0)

x^-5y⁷

x²y^-1

Solution

1

x^-5 y⁷
x² y^-1
2

=

y⁷⋅y¹
x⁵⋅x²
3

=

y^{7 + 1}
x^{5 + 2}
4

=

y⁸
x⁷

1 ~ 2:

x^-5
The exponent is (-).
So move x^-5
from the numerator to the denominator,
and change the exponent -5 to 5.

y^-1
The exponent is (-).
So move y^-1
from the denominator to the numerator,
and change the exponent -1 to 1.

2 ~ 4:

Product of Powers