East Campus, CB 117
361-698-1579
Math Learning Center
West Campus, HS1 203
361-698-1860
EXPONENT RULES
Rule
Description
1 as an
exponent
𝑎𝑎 = 𝑎𝑎1
Product Rule
𝑎𝑎𝑚𝑚 ∙ 𝑎𝑎𝑛𝑛 = 𝑎𝑎𝑚𝑚+𝑛𝑛
0 as an
exponent
Quotient Rule
Power Rule
Raising a
product to a
power
Raising a
quotient to a
power
Negative
exponent
𝑎𝑎0 = 1
Any nonzero number to the power
of zero equals 1.
𝑎𝑎𝑚𝑚
= 𝑎𝑎𝑚𝑚−𝑛𝑛
𝑛𝑛
𝑎𝑎
To divide with the same base,
subtract the exponents.
(numerator) – (denominator)
(𝑎𝑎𝑚𝑚 )𝑛𝑛 = 𝑎𝑎𝑚𝑚∙𝑛𝑛
(𝑎𝑎𝑎𝑎)𝑚𝑚 = 𝑎𝑎𝑚𝑚 𝑏𝑏𝑚𝑚
𝑎𝑎 𝑚𝑚 𝑎𝑎𝑚𝑚
� � = 𝑚𝑚
𝑏𝑏
𝑏𝑏
𝑎𝑎−𝑛𝑛 =
1
𝑎𝑎𝑛𝑛
𝑎𝑎−𝑚𝑚 𝑏𝑏𝑛𝑛
=
𝑏𝑏−𝑛𝑛 𝑎𝑎𝑚𝑚
𝑎𝑎 −𝑛𝑛
𝑏𝑏 𝑛𝑛
� � =� �
𝑏𝑏
𝑎𝑎
Scientific
Notation
Anything “without” a power is
raised to the power of 1.
𝑚𝑚
𝑁𝑁 × 10
(𝐴𝐴 × 10𝑚𝑚 )(𝐵𝐵 × 10𝑛𝑛 ) = 𝐴𝐴𝐴𝐴 × 10𝑚𝑚+𝑛𝑛
(𝐴𝐴 × 10𝑚𝑚 ) ÷ (𝐵𝐵 × 10𝑛𝑛 ) = 𝐴𝐴 ÷ 𝐵𝐵 × 10𝑚𝑚−𝑛𝑛
To multiply with the same bases,
add the exponents.
To raise a power to a power,
multiply the exponents and leave
the base unchanged.
To raise a product to a power,
raise each factor to that power.
To raise a quotient to a power,
raise each numerator and
denominator to the power.
To turn a negative exponent into a
positive exponent, move the base
and exponent to the denominator
(or numerator), depending on
where it is.
In scientific notation there is one
nonzero digit before the decimal
point.
𝐸𝐸𝐸𝐸.
2 × 10−5 𝑜𝑜𝑜𝑜 9.0713 × 106
When multiplying or dividing, be
sure the final answer is in scientific
notation.
East Campus, CB 117
361-698-1579
Math Learning Center
West Campus, HS1 203
361-698-1860
Rule
Example 1
Example 2
1 as an
exponent
5 = 51
0.3 = 0.31
32 ∙ 35 = 32+5 = 37 = 2,187
𝑦𝑦 4 ∙ 𝑦𝑦 5 = 𝑦𝑦 9
0 as an
exponent
Product Rule
Quotient Rule
Power Rule
Raising a
product to a
power
Raising a
quotient to a
power
Negative
exponent
Scientific
Notation
Scientific
Notation
Multiplication
Scientific
Notation
Division
(−2)0 = 1
65
62
5−2
=6
890 = 1
𝑥𝑥 6
= 𝑥𝑥 6−8 = 𝑥𝑥 −2
8
𝑥𝑥
3
= 6 = 216
(𝑧𝑧 4 )3 = 𝑧𝑧 4∙3 = 𝑧𝑧12
(42)5 = 42∙5 = 410 = 1,048,576
(𝑔𝑔ℎ)6 = 𝑔𝑔6 ℎ6
(5𝑥𝑥)3 = 53 𝑥𝑥 3 = 125𝑥𝑥 3
7
1 2 12 1
� � = 2=
3
3
9
−2
5
(𝑥𝑥 3 )7 𝑥𝑥 21
𝑥𝑥 3
= 7
� � =
𝑦𝑦
𝑦𝑦 7
𝑦𝑦
𝑥𝑥 4
𝑥𝑥 4 𝑦𝑦 8
=
= 𝑥𝑥 4 𝑦𝑦 8
−8
𝑦𝑦
1
1
1
= 2=
5
25
. 000014 = 1.4 × 10−5
3,879,000 = 3.879 × 106
(15 × 108 ) ÷ (30 × 105 )
= (15 ÷ 30) × 108−5
= .5 × 103 = 5 × 102
(107.1 × 10−6 ) ÷ (1.05 × 10−2 )
= (107.1 ÷ 1.05) × 10−6−(−2)
= 102 × 10−4 = 1.02 × 10−2
Subtracted one from
the exponent, since
you move the decimal
one spot to the right
Added one to the
exponent, since you
move the decimal
one spot to the left
(−8 × 104 )(4 × 105 )
= (−8 ∙ 4) × 104+5
= −32 × 109 = −3.2 × 1010
Subtracted one from
the exponent, since
you move the decimal
one spot to the right
(. 7 × 10−3 )(. 2 × 10−4 )
= (.7 ∙ .2) × 10−3+(−4)
= .14 × 10−7 = 1.4 × 10−8
Added two to the
exponent, since you
move the decimal two
spots to the left