Download 6.1 Roots and Radical Expressions

6.1 Roots and Radical Expressions
To Simplify:
-Simplify by finding perfect squares (or cubes, etc.)
Examples:
52=25, 5 is a square root of 25 (√25=5)
3
53=125, 5 is a cube root of 125 ( √125=5)
4
54=625, 5 is a fourth root of 625 ( √625=5)
So by definition the nth root: If an=b, then a is the nth root of b.
*VARIABLES:
divide exponent by root to simplify—leave “leftovers” (remainders) inside radical
When root is even, and variable exponent is odd in your answer, put that variable/exponent in
absolute value bars*
Examples:
Leave 1 Line for each set: Find each real root
1a) √36
1b) √0.25
2a)√16𝑥 2
2b) √27𝑦 6
3
3
1c) − √64
4
3
1d) √−27
5
2c) √𝑥 20 𝑦 28 2d) √32𝑦10
6.2 Multiplying and Dividing Radical Expressions
To Multiply:
-Keep numbers outside the radical outside and the numbers inside, inside.
-DO NOT MULTIPLY INSIDE NUMBERS—put them both under radical
(& factor perfect squares--makes simplifying easier)
-Simplify EVERYTHING
To Divide:
-If same root on numerator/denominator, try to divide inside radical first
-Then, simplify all exponents and radicals
(not really “dividing”—just simplifying as much as possible)
-NO radicals can be in denominator
(*rationalize*: multiply top & bottom by a number that will make denominator
a perfect square, cube, etc. so the radical can be simplified to a whole number)
Examples:
Leave 2 lines for each problem: Multiply, then simplify.
1)√8 ∙ √32
3
3
2) √4 ∙ √16
4
4
3) 3√18𝑎9 ∙ √6𝑎𝑏 2
Leave 3 full lines each: Divide, then simplify.
3
√48𝑥 3 𝑦 2
4) 3
√6𝑥 4 𝑦
5)
6)
√𝑥
√2
√5𝑥 4 𝑦
√2𝑥 2 𝑦3
6.3 Binomial Radical Expressions
To Add/Subtract:
-must have like terms/radicals (Similar to like terms w/ variables) *SOMETIMES YOU MUST SIMPLIFY FIRST*
3√3 2√3 -4√3 √3
all like radicals (√3)
3x
2x
-4x
x
all like terms (x)
Example: 4√2+ 5√2 -√2= (4+5-1) √2 = 8√2
…just like 4x + 5x – x = (4+5-1)x = 8x
When Multiplying:
-Remember to FOIL and to multiply things INSIDE radical separate from things OUTSIDE radical
When Dividing:
-Binomial denominator: multiply by CONJUGATE (change middle sign—like we did with i)
**All other exponent rules apply**
6.3a Examples:
Leave 1 line for each problem. Add or subtract if possible.
1) 5√6 + √6
3
2) 4√3 + 4√3
3) 6√18 + 3√50
Leave 2 lines for each problem. Multiply.
4) (2+√7)(1+3√7)
5) (√13 + 6)2
6.3b Examples:
6) (5−√11) (5+√11)
Leave 3 full lines each: Divide. Rationalize.
1+√3𝑥
7)
√6𝑥
8)
5+√3
2−√3
6.4 Rational Exponents
𝑚
𝑛
𝑎 =𝑎
𝑝𝑜𝑤𝑒𝑟
𝑟𝑜𝑜𝑡
𝑛
√𝑎 𝑚
=
𝑎
−𝑚
𝑛
=
𝑚
𝑛
1
𝑝𝑜𝑤𝑒𝑟
𝑎 𝑟𝑜𝑜𝑡
=
𝑛
1
√𝑎𝑚
𝑛
Exponential form: 𝑎
Radical form: √𝑎𝑚
-Multiplying like bases, add exponents; dividing, subtract exponents
-Exponent is outside parentheses: MUTLIPLY exponents
-More than one thing in parentheses: distribute to ALL
-Negative exponents: move to bottom (put under 1 if no numerator)
-All of these work backwards too
*No fractional exponents in bottom of fractions!!
Examples:
Simplify. (1 line)
1a. 27
1
3
1
2
1
2
1b. 7 ∙ 21
Write in radical form. (1 line)
2a. 𝑥
1
3
2b. 𝑥 1.5
Write in exponential form. (1 line)
4
3a. √−10
3b. √𝑐 2
Find each product or quotient. (2 lines)
3
4
4a. √6 ∙ √6
6
4b.
√𝑥𝑦 4
√4
4c. 4
3
√𝑥 8 𝑦 2
√4
Simplify each number. (2 lines)
5a. 8
2
3
3
5
1
3
5b. (−243)
1
3
5c. 27 27
Write in simplest form. (2 lines)
2
3
6a. (𝑥 )−3
1
3
6b. (−27𝑥 −9 )
1
2
−2
3
6c. (𝑥 𝑦 )−6
6d. (
1
𝑥4
−3
𝑦4
12
)
6.5 Solving Square Root and Other Radical Equations
Radical equation: equation that has a variable in a radicand or a variable
with a rational exponent
Extraneous solution: solution that does not work in original equation when
plugged in & checked
Steps to Solve:
1) Isolate radical
2) Square, cube, etc. each side or *multiply fractional exponents by
reciprocal on BOTH sides*
3) Solve for x
[4) **check for extraneous solutions IF variable is on each side of = sign or
you have 2+ variables that don't combine.]
6.5a Examples: (Leave 4-5 lines each)
1) √𝑥 + 4 + 6 = 7
2) √4𝑥 + 1 − 5 = 03)
1
3
3) (6𝑥 + 9) − 5 = −2
5
4) 3 √(𝑥 + 1)3 + 1 = 25
6.5b Examples: (Leave 8-10 lines)
5) √𝑥 + 7 − 5 = 𝑥
6) 2 + √𝑥 − 6 = √𝑥 + 10
6.6 Function Operations
**AICE writes this as: gf(x) which means the same as above**
Examples:
Let f(x) = 3x – 2 and g(x) = x2 + 1. Perform each function operation. (Leave 2 lines each problem)
(f∙g)(x)
(f–g)(x)
(f ° g)(x)
f(x) + g(x)
g(x) – f(x)
f(x) – g(x)
Let f(x) = 2x and h(x) = x2 + 4 (Leave 4 lines each problem)
(f ° h)(1)
(h ° f)(-2)
(f ° h)(a)
Write a formula for fg(x) in each of the following: (Leave 3 lines each problem)
f(x) = 3x
f(x) =
𝑥−3
2
g(x) = x + 2
g(x) = 5x
6.7 Inverse Relations and Functions
Inverse Function (written as f-1)
Steps to find inverse:
1. Change f(x) to y
2. Switch x and y
3. Solve for y
Find the inverse. Is it a function?: (Leave 4 lines each)
1) y = 2x – 1
2) y = (3x – 4)2
3) y =
2𝑥
5
Graph each relation and its inverse. (Leave room for a graph and 3 lines for work)
4) y = 2x + 8
If f(x) = x + 2, evaluate: (Leave 4 lines)
5) f-1(3)
If g(x) = 2(x – 1), evaluate: (Leave 4 lines)
6) g-1(2)