7.5 – Radical Expressions. Radical Radical – radical symbol.

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Presentation transcript:

7.5 – Radical Expressions

Radical

Radical – radical symbol

Radical – radical symbol =

Recall:

Radical – radical symbol = Recall: Ex. 1 Simplify

Radical – radical symbol = Recall: Ex. 1 Simplify = ____ √ 3·3·3

Radical – radical symbol = Recall: Ex. 1 Simplify = 2 ____ √ 3·3·3

Radical – radical symbol = Recall: Ex. 1 Simplify = 2 ____ √ 3·3·3

Radical – radical symbol = Recall: Ex. 1 Simplify = 2 √ 3·3·3 √ 3·3 · √ 3

Radical – radical symbol = Recall: Ex. 1 Simplify = 2 √ 3·3·3 √ 3·3 · √ 3 3 · √ 3

Radical – radical symbol = Recall: Ex. 1 Simplify = 2 √ 3·3·3 √ 3·3 · √ 3 3 √ 3

Ex. 2 Simplify √ 16p 8 q 7

√ 16p 8 q 7 =

Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16

Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16 · √ p 8

Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16 · √ p 8 · √ q 7

Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16 · √ p 8 · √ q 7 = √4 2

Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16 · √ p 8 · √ q 7 = √4 2 ·√(p 4 ) 2

Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16 · √ p 8 · √ q 7 = √4 2 ·√(p 4 ) 2 ·√(q 6 )q

Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16 · √ p 8 · √ q 7 = √4 2 ·√(p 4 ) 2 ·√(q 6 )q = √4 2 ·√(p 4 ) 2 ·√(q 3 ) 2 ·q

Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16 · √ p 8 · √ q 7 = √4 2 ·√(p 4 ) 2 ·√(q 6 )q = √4 2 ·√(p 4 ) 2 ·√(q 3 ) 2 ·q = √4 2 ·√(p 4 ) 2 ·√(q 3 ) 2 ·√q

Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16 · √ p 8 · √ q 7 = √4 2 ·√(p 4 ) 2 ·√(q 6 )q = √4 2 ·√(p 4 ) 2 ·√(q 3 ) 2 ·q = √4 2 ·√(p 4 ) 2 ·√(q 3 ) 2 ·√q = 4 · p 4 · q 3 ·√q

Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16 · √ p 8 · √ q 7 = √4 2 ·√(p 4 ) 2 ·√(q 6 )q = √4 2 ·√(p 4 ) 2 ·√(q 3 ) 2 ·q = √4 2 ·√(p 4 ) 2 ·√(q 3 ) 2 ·√q = 4 · p 4 · q 3 ·√q = 4p 4 q 3 √q

Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 33

Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 33

Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 33

Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 33

Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 33 3

Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n

Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n

Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n

Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 18 · √ 3 2 ·n 2 ·3·2 3 ·n

Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 18 · √ 3 2 ·n 2 ·3·2 3 ·n

Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 18 · √ 3 2 ·n 2 ·3·2 3 ·n 18 · √ 3 3 ·n 3 ·

Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 18 · √ 3 2 ·n 2 ·3·2 3 ·n 18 · √ 3 3 ·n 3 ·

Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 18 · √ 3 2 ·n 2 ·3·2 3 ·n 18 · √ 3 3 ·n 3 ·

Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 18 · √ 3 2 ·n 2 ·3·2 3 ·n 18 · √ 3 3 ·n 3 · · √ 3 3 ·√ n 3 ·√

Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 18 · √ 3 2 ·n 2 ·3·2 3 ·n 18 · √ 3 3 ·n 3 · · √ 3 3 ·√ n 3 ·√ · 3 · n ·

Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 18 · √ 3 2 ·n 2 ·3·2 3 ·n 18 · √ 3 3 ·n 3 · · √ 3 3 ·√ n 3 ·√ · 3 · n · 2 18 · 3 · 2 · n

Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 18 · √ 3 2 ·n 2 ·3·2 3 ·n 18 · √ 3 3 ·n 3 · · √ 3 3 ·√ n 3 ·√ · 3 · n · 2 18 · 3 · 2 · n 108n

Ex. 4 Simplify each expression. a. 2√ √ √ 48

Ex. 4 Simplify each expression. a. 2√ √ √ 48 2√ 2·2·3 - 3√ 3·3·3 + 2√ 2·2·2·2·3

Ex. 4 Simplify each expression. a. 2√ √ √ 48 2√ 2·2·3 - 3√ 3·3·3 + 2√ 2·2·2·2·3 2√ 2 2 ·3 - 3√ √ 2 4 ·3

Ex. 4 Simplify each expression. a. 2√ √ √ 48 2√ 2·2·3 - 3√ 3·3·3 + 2√ 2·2·2·2·3 2√ 2 2 ·3 - 3√ √ 2 4 ·3 2√ 2 2 ·3 - 3√ √ 2 2 ·2 2 ·3

Ex. 4 Simplify each expression. a. 2√ √ √ 48 2√ 2·2·3 - 3√ 3·3·3 + 2√ 2·2·2·2·3 2√ 2 2 ·3 - 3√ √ 2 4 ·3 2√ 2 2 ·3 - 3√ √ 2 2 ·2 2 ·3 2·2·√3 - 3·3·√3 + 2 ·2·2· √3

Ex. 4 Simplify each expression. a. 2√ √ √ 48 2√ 2·2·3 - 3√ 3·3·3 + 2√ 2·2·2·2·3 2√ 2 2 ·3 - 3√ √ 2 4 ·3 2√ 2 2 ·3 - 3√ √ 2 2 ·2 2 ·3 2·2·√3 - 3·3·√3 + 2 ·2·2· √3 4√ 3 - 9√ 3 + 8√ 3

Ex. 4 Simplify each expression. a. 2√ √ √ 48 2√ 2·2·3 - 3√ 3·3·3 + 2√ 2·2·2·2·3 2√ 2 2 ·3 - 3√ √ 2 4 ·3 2√ 2 2 ·3 - 3√ √ 2 2 ·2 2 ·3 2·2·√3 - 3·3·√3 + 2 ·2·2· √3 4√ 3 - 9√ 3 + 8√ 3

Ex. 4 Simplify each expression. a. 2√ √ √ 48 2√ 2·2·3 - 3√ 3·3·3 + 2√ 2·2·2·2·3 2√ 2 2 ·3 - 3√ √ 2 4 ·3 2√ 2 2 ·3 - 3√ √ 2 2 ·2 2 ·3 2·2·√3 - 3·3·√3 + 2 ·2·2· √3 4√ 3 - 9√ 3 + 8√ 3 -5√ 3 + 8√ 3

Ex. 4 Simplify each expression. a. 2√ √ √ 48 2√ 2·2·3 - 3√ 3·3·3 + 2√ 2·2·2·2·3 2√ 2 2 ·3 - 3√ √ 2 4 ·3 2√ 2 2 ·3 - 3√ √ 2 2 ·2 2 ·3 2·2·√3 - 3·3·√3 + 2 ·2·2· √3 4√ 3 - 9√ 3 + 8√ 3 -5√ 3 + 8√ 3

Ex. 4 Simplify each expression. a. 2√ √ √ 48 2√ 2·2·3 - 3√ 3·3·3 + 2√ 2·2·2·2·3 2√ 2 2 ·3 - 3√ √ 2 4 ·3 2√ 2 2 ·3 - 3√ √ 2 2 ·2 2 ·3 2·2·√3 - 3·3·√3 + 2 ·2·2· √3 4√ 3 - 9√ 3 + 8√ 3 -5√ 3 + 8√ 3 3√ 3

b. (5√ 3 – 6)(5√ 3 +6)

5√ 3(5√ 3 +6)

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6)

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6)

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6)

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6 25√ √ 3 – 30√ 3 – 36

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6 25√ √ 3 – 30√ 3 – 36

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6 25√ √ 3 – 30√ 3 – 36 25·3

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6 25√ √ 3 – 30√ 3 – 36 25·3

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6 25√ √ 3 – 30√ 3 – 36 25·3+ 0 – 36

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6 25√ √ 3 – 30√ 3 – 36 25·3 – – 36

b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6 25√ √ 3 – 30√ 3 – 36 25·3 – – 36 39