### Math 102 Lecture Notes Ch. 4.2 Page 1 of 7 4.2 Simplifying Radicals

```Math 102 Lecture Notes Ch. 4.2 4.2 Simplifying Radicals We say that a square root radical expression is in simplest form when 1. its radicand has no square factors 2. the radicand is not a fraction 3. no denominator contains a radical Example (a) Which of the following radicals are in simplest form? (i) 33 (iii) (ii) 20 33 = 3 •11 20 = 4 • 5 The radicand has no square factors, so the radical 4 is a square factor of the radicand, so the radical is in simplest form. 1
3
(iv) 1
5
The radicand is a fraction, so the radical The denominator contains a radical, so the expression is not in simplest form. is not in simplest form. is not in simplest form. To simplify square root radicals we can use the following rules of square roots: Product Rule ab = a • b (for a ≥ 0 and b ≥ 0) Example (b) Simplify 20 . 20 = 2 2 • 5 = 2 2 • 5 = 2 • 5 = 2 5 Example (c) Simplify 180 . 180 = 2 2 • 32 • 5 = 2 2 • 32 • 5 = 2 • 3 • 5 = 6 5 Page 1 of 7 Quotient Rule a
a
=
(for a ≥ 0 and b > 0) b
b
Math 102 Lecture Notes Demonstration Problems Simplify Practice Problems Simplify 1. (a) 28 2. (a) 50 3. (a) 300 150
4. (a) 49 1. (b) 40 2. (b) 75 3. (b) 500 175
4. (b) 9
Ch. 4.2 Page 2 of 7 10 ; 2. (b) 5 3 ; 3. (b) 10 5 ; 4. (b) 5 7
3
Math 102 Lecture Notes Ch. 4.2 In general, an nth root radical is considered to be in simplest form when 1. its radicand has no factors raised to a power greater than or equal to the index 2. the radicand is not a fraction 3. no denominator contains a radical Example (d) Which of the following radicals are in simplest form? (i) 3 18 3
18 = 3 2 • 32 4
The radicand has no cube factors, so the radical 48 = 4 2 4 • 3 24 is 4th power factor in the radicand, so the radical is in simplest form. 1
4
(iii) 5
(ii) 4 48 (iv) 5
5
4
7
3
The radicand is a fraction, so the radical The denominator contains a radical, so the radical is not in simplest form. is not in simplest form. is not in simplest form. To simplify nth root radicals we can use the following rules of nth roots: Product Rule n
Quotient Rule ab = n a • n b n
(for a ≥ 0 and b ≥ 0 when n is even) (for a ≥ 0 and b > 0 when n is even) Example (e) Simplify 4 48 . 48 = 4 2 4 • 3 = 4 2 4 • 4 3 = 2 • 4 3 = 2
4
4
3 Example (f ) Simplify 5 −64 . −64 = 5 −1• 2 6 = 5 −1• 2 5 • 2 = 5 −1 • 5 2 5 • 5 2 = −1• 2 • 5 2 = −2
5
a na
=
b nb
5
2 Page 3 of 7 Math 102 Lecture Notes Ch. 4.2 Demonstration Problems Simplify 5. (a) 3 40 6. (a) 3 −81 Practice Problems Simplify 5. (b) 3 54 6. (b) 3 −250 7. (b) 4 162 8. (b) − 5 160 7. (a) 4 112 8. (a) − 5 96 Answers: 5. (b) 3 3 2 ; 6. (b) −5 3 2 ; 7. (b) 3 4 2 ; 8. (b) −2 5 5 Page 4 of 7 Math 102 Lecture Notes Ch. 4.2 Variable Radical Expressions To simplify a radical that contains variables, we continue to use the rules: a 2 = a (for a ≥ 0) n
a n = a (for a ≥ 0 when n is even) n
ab = n a • n b (for a ≥ 0 and b ≥ 0 when n is even) n
a na
=
b nb
(for a ≥ 0 and b > 0 when n is even) In all of the exercises in this chapter, assume all variables represent positive real numbers. Example (g) Simplify 8y 3 . 8y 3 = 2 3 y 3 = 2 2 • 2 • y 2 • y = 2 2 • y 2 • 2 • y = 2 2 • y 2 • 2 • y = 2 • y • 2 • y = 2y 2y Example (h) Simplify 3 x 8 . x 8 = 3 x 3 • x 3 • x 2 = 3 x 3 • 3 x 3 • 3 x 2 = x • x • 3 x 2 = x
Alternative method: =
3
=
3
= x 2
3
3
2 3
x x 8 = 3 x 6 • x 2 (x )
• x 2 (x )
• 3 x 2 2 3
2 3
3
2
x 2 Page 5 of 7 Math 102 Lecture Notes Example (i) Simplify Ch. 4.2 9a 2
. 49b 2
9a 2
=
49b 2
=
=
=
9a 2
49b 2
32 a 2
72 b2
32 • a 2
72 • b2
3a
7b
Demonstration Problems Simplify 9. (a) 36a b 25
10. (a) 2 x
2 2
Practice Problems Simplify 9. (b) 4x 4 8
10. (b) 3 3 x
2
Answers: 9. (b) 2x2; 10. (b) x ; Page 6 of 7 Math 102 Lecture Notes Ch. 4.2 Demonstration Problems Simplify Simplify 11. (a) 27x 11. (b) 3 8a 4 12. (a) 40x 5 y 6 z 8 12. (b) 90x 4 y 5 z 6 3
13. (a) 4
5
a5
16b 8
Practice Problems 13. (b) 3
a6
27b12
Answer: 11. (b) 2a 3 a ; 12. (b) 3x 2 y 2 z 3
Page 7 of 7 10y ; 13. (b) a2
3b 4
```