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Exponent

Summary: Exponent Rules

Product of Powers

Formula

aman = am + n

Example

Simplify the given expression.

x6⋅x8

Solution

  • 1
    x6x8 = x6 + 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 y43 x3 y7
  • 2
    32 + 1x1 + 3y4 + 7
  • 3
    = 33⋅x4⋅y11
  • 4
    = 27x4y11
1 ~ 2:
Multiply the same base numbers.

Power of a Power

Formula

(am)n = amn

Example

Simplify the given expression.

(x2)3

Solution

  • 1
    (x2)3 = x23
  • 2
    = x6

Example

Simplify the given expression.

((x4)2)5

Solution

  • 1
    ((x4)2)5
  • 2
    = (x42)5
  • 3
    (x8)5
  • 4
    = x85
  • 5
    = x40
1 ~ 2:
Solve the inner power.
3 ~ 4:
Then solve the outer power.

Power of a Product

Formula

(ab)m = am⋅bm

Example

Simplify the given expression.

(x3y)2

Solution

  • 1
    (x3y)2
  • 2
    = (x3)2⋅y2
  • 3
    = x6y2
2 ~ 3:
(x3)2
= x3⋅2
= x6

Power of a Power

Quotient of Powers

Formula

am
an
  = am - n

Example

Simplify the given expression. (x ≠ 0)

x8
x3

Solution

  • 1
    x8
    x3
     = x8 - 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
    x4 y2 z5
  • 2
    = 3⋅x7 - 4z9 - 5
  • 3
    = 3x3z4
1 ~ 2:
12/3 = 4
And cancel y2.

Power of a Quotient

Formula

(
a
b
)
m
 = 
am
bm

Example

Simplify the given expression. (y ≠ 0)

(
x
y2
)
3

Solution

  • 1
    (
    x
    y2
    )
    3
  • 2
    x2
    (y2)3
  • 3
    x2
    y6
2 ~ 3:
(y2)3
= y2⋅3
= y6

Power of a Power

Example

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

(
3x4
y
)
2

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
= x4⋅2
= x8

Power of a Power
3 ~ 4:
x8/x3
= x8 - 3
= x5

y3/y2
= y3 - 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
    = 4x211
  • 3
    = 4x2

Negative Exponent

Formula

a-m = 
1
am

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)

1
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
    y7y1
    x5⋅x2
  • 3
    y7 + 1
    x5 + 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.