radical functions & rational exponents unit objectives: simplify radical and rational exponent...
TRANSCRIPT
Radical Functions & Rational ExponentsUnit Objectives:
• Simplify radical and rational exponent expressions• Solve radical equations• Find the inverse of a function: algebraically & graphically• Identify function attributes: domain and range• Perform function operations: add, subtract, multiply & compose
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
=¿𝑎𝑛
𝑏𝑛
1
𝑏𝑛𝑏−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=¿53√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
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