6-1 Radical Functions & Rational Exponents

Unit Objectives:• Simplify radical and rational exponent expressions• Solve radical equations• Solve rational exponent equations

Today’s Objective:I can simplify radical expressions.

Review of Exponent Properties𝑏𝑚 ⋅𝑏𝑛=𝑏❑❑𝑚+𝑛 (𝑏¿¿𝑚)𝑛=𝑏❑ ¿❑𝑚⋅𝑛

𝑏𝑚

𝑏𝑛 =𝑏❑

❑𝑚−𝑛

(𝑎𝑏)𝑛=¿𝑎𝑛 ⋅𝑏𝑛

(𝑎𝑏 )𝑛=¿𝑎

𝑛

𝑏𝑛1𝑏𝑛𝑏−𝑛=¿

1𝑏0=¿

Simplify with positive exponents only.

9𝑥10

𝑦6

2 𝑥5 ⋅3 𝑥8 (3 𝑥5 𝑦−3)2 ( 3 𝑥− 2 𝑦3

𝑥5 𝑦7 )− 1

6 𝑥13 𝑥7 𝑦 4

3

Roots & Radical ExpressionsPowers Roots Radicals

22=¿4 2 is the square root of 4 ❑√ 4=¿❑√22=¿223=¿8 2 is the cube root of 8 3√8=¿3√23=¿224=¿16 2 is the fourth root of 16 4√16=¿4√24=¿2𝑎𝑛=¿𝑏 a is the nth root of b 𝑛√𝑏=¿𝑛√𝑎𝑛=¿𝑎

bnIndex:

Degree of root

Radicand

3√(−3)3=¿−3

3√53=¿

Simplifying Radicals𝑛√𝑎𝑛={ 𝑎 ,if 𝑛is odd|𝑎|, if 𝑛 is even

1. Write radicand in factors raised to the nth power or less.

2. Take the nth root of all factors to the nth power.3. Simplify in front of radical and under radical.

❑√25=¿❑√52=¿ |3√125=¿ 5

3√−27=¿3√64=¿

2 ⋅2=¿3√8 ⋅ 8

¿ 3√23 ⋅23=¿ 4

3√𝑥12=¿¿ 𝑥⋅ 𝑥⋅ 𝑥 ⋅ 𝑥¿ 𝑥4

5√32𝑥10=¿¿2 ⋅ 𝑥 ⋅ 𝑥¿2 𝑥2

3√𝑥3⋅ 𝑥3⋅ 𝑥3⋅ 𝑥3

5√25 ⋅ 𝑥5 ⋅ 𝑥5

¿5

Simplifying Radicals𝑛√𝑎𝑛={ 𝑎 ,if 𝑛is odd|𝑎|, if 𝑛 is even

1. Write radicand in factors raised to the nth power or less.

2. Take the nth root of all factors to the nth power.3. Simplify in front of radical and under radical.

❑√24=¿2❑√6

❑√ 4 ⋅6❑√22 ⋅6=¿

3√24=¿2 3√3

3√8 ⋅33√23 ⋅3=¿

3√54 𝑥5=¿3√27 ⋅2 ⋅ 𝑥3⋅ 𝑥2

¿3 𝑥 3√2 𝑥2

4√ 48𝑥13=¿4√16 ⋅3 ⋅ 𝑥4 ⋅ 𝑥4⋅ 𝑥4 ⋅ 𝑥

¿2 ⋅∨𝑥∨⋅∨𝑥∨⋅∨𝑥∨⋅ 4√3 𝑥¿2¿ 𝑥3∨4√3 𝑥

¿ 3√33⋅2 ⋅ 𝑥3⋅ 𝑥2

¿ 4√24⋅ 3 ⋅𝑥4 ⋅ 𝑥4 ⋅ 𝑥4 ⋅𝑥