Honors Algebra II 0809

You can use exponents and roots to solve equations…

1. x^2=16

√x2=√16

x=±4

2. x^3=27

3√x3= 3√27

x=3

What about when there is more complication???

1.  2x^4=16

divide by 2.

 2 4√x4= 4√81.

x = ±3

2. 3√(x-3)^3= 3√10

the threes cancell out to equal one

x-3= 3√10

x = 3√10 +3

1.5√4x^5= 5√-17

4x = 5√-17

2. 3(X^4 – 19)=9999

3X64 – 57 =9999

X^4 – 19 =3333

4√X^4 = 4√3352

X= ± 4√3352

3. 3(x + 3)^10 + 20=-100

2(x + 3)^10/ (x + 3)^10 =-120/-60

10√(x + 3)^10= 10√-60

x + 3 =± 10√-60 NO SOLUTION

HOMEWORK

PG. 404 8-11, 56-58

PG. 411 (22-40 EVENS) 56-63

There are many rules to exponents; however we have used them many times over by now.

a^m * a^n = a^m+n

(ab)^m = a^m*b^m

a^m/b^n = a^ m-n

(a^m)^n = a^mn

a^-m = 1/a^m

(a/b)^m = a^m/b^m

Examples Involving Exponents/ Roots

5^1/2 * 5^1/4 = 5^1/2 + 1/4 = 5^3/4

(8^1/2 * 5^1/3)^2 = (8^1/2)^ 2 * (5^1/3) = 8 * 5^1/3

(2^4 * 81)^ -1/4 = 2^-1 * 81^-1/4 = 1/2 * 1/81^1/4 = 1/2 * 1/3 = 1/6

More Examples Involving Exponents/ Roots

4^√16/81 = 4^√16/ 4^√81 = 2/3

3^√ x^3 / z = x/ z^1/3 or x/ 3^√z

x * 3^√y^6 + y^2 * 3^√ x^3 = xy^6/3 + y^2 x = xy^2 + y^2 + x = 2xy^2

Even More Examples…

70^1/3 / 14^1/3 = (70/14)^1/3 = 5^1/3 = 3^√5

4^√9 * 4^√6 = 4^√9 * 6 = 4^√ 54

(6^√8 * 6 √ 16) / 6^√2 = 6^√ (8 * 16) / 2 = 6^√ 128 / 2 = 6^√64 = 2

Homework

page 441 # 5 – 55 ODD

In general ⁿ√x represents the “nth root of X” (normally we take the SQUARE ROOT)

      √9 = ²√9 = 3     because…   3*3 = 9

Roots besides SQUARE ROOT

      ³√X   “third root of X” NOT “third SQUARE root of X”

        ³√27 = 3           n*n*n = 27

alternative way of writing these exponents

       ³√X = X^1/3  (the reciprical)

         ³√X²  (if squared then…) = X^2/3  (the exponent goes on TOP of the fraction)

ex:

       ⁴√2⁴  = 2^{4/4  (4/4=1)}  = 2

√16 = 4                                                       7776^1/5  = 6

253^1/3 = 5                                                   ⁶√1 = 1

⁴√256 = 4                                                    128^1/7 = 2
                                                                                                                               

(-16)^1/2 = NP (not possible)                          ⁵√-7776 = -6

³√-125 = -5                                                (-1)^1/6 = NP

(-256)^1/4 = NP                                                         ⁷√-128 = -2

(when negative and has an even exponent or root, it equals Not Possible)

combining powers with roots…

             X^m/n  = ⁿ√X^m                       3^5/7  = ⁷ √3⁵

                 2^-5/3  = ³√2^-5 = ³√1/2⁵ = ³√1/32
                                                                                                                                 

example problems:

1.) (³√-27)^-4 = -27^{-4 /3} = -1/81

2.) (³√-8)⁵ = -8^{5 /3} = -32

3.) (-125)^-2/3 = -125^{-2 /3} = -1/25

HOMEWORK: pg 404 #4-7, (13-21 odds), (30-46 evens)   

Equation: y=(.5x)^2

vertex: (0,0)

shift/streches: 2 times as wide

Parabola                         quadratic

                                         y=ax2+bx+C

Don’t like graphing?

Just say one thing to yourself…..Mother Function!

general vertex form     y=a(x-h)2 + k

*All other parabolas are based off this equation

y=2x^2–2 times as tall

y=.5x^2–.5 times as tall

y=a(x-h)^2+k

a=height factor

h=shift left/right

k=shift up/down “elevator”