NAME _____________________________________________ DATE ____________________________ PERIOD _____________
6-5 Study Guide and Intervention
Operations with Radical Expressions
Simplify Radicals
Product Property of Radicals
For any real numbers a and b, and any integer n > 1:
𝑛
𝑛
𝑛
1. if n is even and a and b are both nonnegative, then √𝑎𝑏 = √𝑎 ⋅ √𝑏.
𝑛
𝑛
𝑛
2. if n is odd, then √𝑎𝑏 = √𝑎 ⋅ √𝑏.
To simplify a square root, follow these steps:
1. Factor the radicand into as many squares as possible.
2. Use the Product Property to isolate the perfect squares.
3. Simplify each radical.
For any real numbers a and b ≠ 0, and any integer n > 1,
Quotient Property of Radicals
𝑛 𝑎
√𝑏 =
𝑛
√𝑎
𝑛
√𝑏
, if all roots are defined.
To eliminate radicals from a denominator or fractions from a radicand, multiply the numerator and denominator by a
quantity so that the radicand has an exact root.
𝟑
Example 1: Simplify √−𝟔𝒂𝟓 𝒃𝟕.
Example 2: Simplify √
3
√−16𝑎5 𝑏 7 = √(−2)3 ⋅ 2 ⋅ 𝑎3 ⋅ 𝑎2 ⋅ (𝑏 2 )3 ⋅ 𝑏
3
= – 2a𝑏 2 √2𝑎2 𝑏
8𝑥 3
8𝑥 3
√45𝑦5 = √45𝑦5
=
=
𝟖𝒙𝟑
𝟒𝟓𝒚𝟓
Quotient Property
√(2𝑥)2 ⋅ 2𝑥
√(3𝑦 2 )2 ⋅ 5𝑦
√(2𝑥)2 ⋅ √2𝑥
√(3𝑦 2 )2 ⋅ √5𝑦
=
2|𝑥|√2𝑥
3𝑦 2 √5𝑦
=
2|𝑥|√2𝑥
3𝑦 2 √5𝑦
=
2|𝑥|√10𝑥𝑦
15𝑦 3
Factor into squares.
Product Property
Simplify.
⋅
√5𝑦
√5𝑦
Rationalize the denominator.
Simplify.
Exercises
Simplify.
1. 5√54
36
4.√125
4
2.√32𝑎9 𝑏 20
5.√
𝑎 6 𝑏3
98
3. √75𝑥 4 𝑦 7
3
6. √
𝑝5 𝑞 3
40
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
6-5 Study Guide and Intervention (continued)
Operations with Radical Expressions
Operations with Radicals When you add expressions containing radicals, you can add only like terms or like radical
expressions. Two radical expressions are called like radical expressions if both the indices and the radicands are alike.
To multiply radicals, use the Product and Quotient Properties. For products of the form (a√𝑏 + c √𝑑) ⋅ (e √𝑓 + g √ℎ), use
the FOIL method. To rationalize denominators, use conjugates. Numbers of the form a √𝑏 + c √𝑑 and a√𝑏 – c √𝑑, where
a, b, c, and d are rational numbers, are called conjugates. The product of conjugates is always a
rational number.
Example 1: Simplify 2√𝟓𝟎 + 𝟒√𝟓𝟎𝟎 – 𝟔√𝟏𝟐𝟓.
2√50 + 4√500 – 6√125 = 2√52 ⋅ 2 + 4√102 ⋅ 5 – 6√52 ⋅ 5
Factor using squares.
= 2 ⋅ 5 ⋅ √2 + 4 ⋅ 10 ⋅ √5 – 6 ⋅ 5 ⋅ √5
Simplify square roots.
= 10√2 + 40 + √5 – 30√5
Multiply.
= 10√2 + 10√5
Combine like radicals.
Example 2: Simplify (2√𝟑 – 4√𝟐 ) (√𝟑+ 2√𝟐) .
Example 3: Simplify
(2√3 – 4√2 ) (√3 + 2√2)
2 − √5
3 + √5
= 2 √3 ⋅ √3 + 2√3 ⋅ 2√2 – 4√2 ⋅ √3 – 4√2 ⋅ 2√2
= 6 + 4√6 – 4√6 – 16
𝟑 + √𝟓
=
2 − √5
3 + √5
=
6 − 2√5 − 3√5 + (√5)2
32 − (√5)2
=
6 − 5√5 + 5
9−5
=
11− 5√5
4
= –10
⋅
3 − √5
3− √5
𝟐 − √𝟓
.
Exercises
Simplify.
1. 𝟑√2 + √50 – 4√8
3
3
2. √20 + √125 – √45
3
3
3
3. √300 – √27 – √75
4.√81 ⋅√24
5. √2(√4 + √12)
6. 2√3 (√15 + √60)
7. (2 + 3√7) (4 + √7)
8. (6√3 – 4√2) (3√3 + √2)
9. (4√2 – 3√5) (2√20 + 5)
10.
5√48 + √75
5√3
4 + √2
√2
11. 2 −
5 + 3√3
√3
12. 1 − 2
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
6-7 Skills Practice
Solving Radical Equations and Inequalities
Solve each equation.
1. √𝑥 = 5
2. √𝑥 + 3 = 7
3. 5√𝑗 = 1
4. 𝑣 2 + 1 = 0
1
1
𝟑
5. 18 − 3𝑦 2 = 25
6. √2𝑤 = 4
7. √𝑏 − 5 = 4
8. √3𝑛 + 1 = 5
𝟑
9. √3𝑟 − 6 = 3
11. √𝑘 − 4 – 1 = 5
1
10. 2 + √3𝑝 + 7 = 6
1
12. (2𝑑 + 3)3 = 2
1
13. (𝑡 − 3)3 = 2
14. 4 – (1 − 7𝑢)3 = 0
15. √3𝑧 − 2 = √𝑧 − 4
16. √𝑔 + 1 = √2𝑔 − 7