Exponent
Summary: Exponent Rules
Product of Powers
Formula
am⋅an = am + n
Example
Simplify the given expression.
x6⋅x8
x6⋅x8
Solution
-
1x6⋅x8 = x6 + 8
-
2= x14
x6 and x8 have the same base: x.
So use the product of powers rule.
So use the product of powers rule.
Example
Simplify the given expression.
32xy4⋅3x3y7
32xy4⋅3x3y7
Solution
-
132 x y4⋅3 x3 y7
-
2= 32 + 1⋅x1 + 3⋅y4 + 7
-
3= 33⋅x4⋅y11
-
4= 27x4y11
1 ~ 2:
Multiply the same base numbers.
Power of a Power
Formula
(am)n = am⋅n
Example
Simplify the given expression.
(x2)3
(x2)3
Solution
-
1(x2)3 = x2⋅3
-
2= x6
Example
Simplify the given expression.
((x4)2)5
((x4)2)5
Solution
-
1((x4)2)5
-
2= (x4⋅2)5
-
3= (x8)5
-
4= x8⋅5
-
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
(x3y)2
Quotient of Powers
Formula
am
an
= am - n
Example
Simplify the given expression. (x ≠ 0)
x8
x3
Solution
-
1x8x3= x8 - 3
-
2= x5
x8 and x3 have the same base: x.
So use the quotient of powers rule.
So use the quotient of powers rule.
Example
Simplify the given expression. (x, y, z ≠ 0)
12x7y2z9
4x4y2z5
Solution
-
112 x7 y2 z94 x4 y2 z5
-
2= 3⋅x7 - 4⋅z9 - 5
-
3= 3x3z4
1 ~ 2:
12/3 = 4
And cancel y2.
And cancel y2.
Power of a Quotient
Formula
(
a
b
)
=
am
bm
Example
Simplify the given expression. (y ≠ 0)
3
(
x
y2
)
Example
Simplify the given expression. (x, y ≠ 0)
2
(
3x4
y
)
Solution
-
1(3x4y2)⋅(y2x3)
-
2=32⋅(x4)2y2⋅y323⋅x3
-
3=9⋅x8y2⋅y38⋅x3
-
4=9x5y8
2 ~ 3:
3 ~ 4:
Zero Exponent
Formula
a0 = 1
Example
Simplify the given expression.
20
20
Solution
-
120 = 1
Example
Simplify the given expression. (x, y ≠ 0)
4x2y0⋅(-3xy)0
4x2y0⋅(-3xy)0
Solution
-
14x2 y0⋅(-3xy)0
-
2= 4x2⋅1⋅1
-
3= 4x2
Negative Exponent
Formula
a-m =
1
am
Example
Simplify the given expression. (x ≠ 0)
x-4
x-4
Solution
-
1x-4 =1x4
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.
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
-
11x-3=x31
-
2= x3
x-3
The exponent is (-).
So move x-3
from the denominator to the numerator,
and change the exponent -3 to 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
-
1x-5 y7x2 y-1
-
2=y7⋅y1x5⋅x2
-
3=y7 + 1x5 + 2
-
4=y8x7
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.
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: