Exponent and Radical Rules
Adding with exponents: You can only add exponents if they have like terms.
Example: 4𝑥 2 + 5𝑥 2 = 9𝑥 2
4𝑥 2 + 5𝑥 3 = 4𝑥 2 + 5𝑥 3
Subtracting with exponents: You can only subtract exponents if they have like terms.
Example: 9𝑚2 − 7𝑚2 = 2𝑚2
9𝑚2 − 7𝑚 = 9𝑚2 − 7𝑚
Multiplying with exponents: When multiplying, combine the terms and add the exponents.
Example: 2𝑦 2 × 2𝑥 2 = 2𝑥𝑦 4
Multiplying with parentheses: 1st Step—Multiply through the parentheses.
Example: (2𝑥 + 2𝑦) (𝑥 + 𝑦) = 2𝑥 × 𝑥 = 2𝑥 2
2𝑥 × 𝑦 = 2𝑥𝑦
2𝑦 × 𝑥 = 2𝑥𝑦
2𝑦 × 𝑦 = 2𝑦 2
Or
2𝑥 2 + 2𝑥𝑦 + 2𝑥𝑦 + 2𝑦 2
2nd Step—Add all like terms
Example: 2𝑥 2 + 2𝑥𝑦 + 2𝑥𝑦 + 2𝑦 2 = 2𝑥 2 + 4𝑥𝑦 + 2𝑦 2
Dividing with exponents: When dividing, divide the numbers and then subtract the exponents.
Example: 4𝑤 3 ÷ 2𝑤 = 4 ÷ 2 = 2
𝑤3 − 𝑤 = 𝑤2
Or
4𝑤 3 ÷ 2𝑤 = 2𝑤 2
Square roots with exponents: Always, find the square of the number and then multiply the
1
exponent by 2.
4
Example: √814 = 9 × 9
1
1
Or
4
×2 =
1
×
1
=
2
√814 = 92
2
1
=2
1
Square roots with exponents: If you cannot divide the exponent by 2 leave it inside of the
radical symbol.
5
Example: √25𝑦 5 = 5 × 5
×
1
Or
1
2
≠ (exponents cannot equal a fraction)
�25 𝑦 5 = 5�𝑦 5
Cubed roots with exponents: Always, find the cubed root of the number and then multiply the
1
exponent by 3.
6
3
Example: √646 = 4 × 4 × 4
1
Or
1
×3 =
6
1
×
1
3
2
=1=2
3
√646 = 42
1
Cubed roots with exponents: If you cannot divide the exponent by 3 leave it inside of the
radical symbol.
1
3
Example: √125𝑥 = 5 × 5 × 5
3
1
Or
3
√125𝑥 = 5 √𝑥
×
1
3
≠ (exponents cannot equal a fraction)
Multiplying with even and odd exponents.
−𝒙𝟐 = +
Example: -22 =
−2 × (−2) = 4
Example: −23 =
−2 × (−2) × (−2) = −8
−𝒙𝟑 = −
(If the number is negative: even exponents will produce a positive answer and odd exponents will
produce a negative answer).