Aim: Exponential Equations Course: Alg. 2 & Trig. Aim: How do we solve equations with fractions, negative numbers, or variables in the exponents? Do Now:

$Page 1: Aim: Exponential Equations Course: Alg. 2 & Trig. Aim: How do we solve equations with fractions, negative numbers, or variables in the exponents? Do Now:$
Aim: Exponential Equations Course: Alg. 2 & Trig.

Aim: How do we solve equations with fractions, negative numbers, or variables in the exponents?

Do Now:

Meteorologists use the formula D3 = 216T2 to describe the size and duration of storms. In the formula, D is the diameter of the storm in miles and T is the duration, or the number of hours the storm lasts. If the diameter of the thunderstorm is 12 miles, about how long would this storm last?


Model Problem

Meteorologists use the formula D3 = 216T2 to describe the size and duration of storms. In the formula, D is the diameter of the storm in miles and T is the duration, or the number of hours the storm lasts. If the diameter of the thunderstorm is 12 miles, about how long would this storm last?

D3 = 216T2

123 = 216T2substitute D = 12

1728 = 216T2simplify and solve

8 = T2

28 T

T 2.8 hours

square root of both sides


Power of Power Rule

Power of Power Property (am)n = am•n

(x2)1/2 =

(x -1/2)-2 = x1 = x

How do we use this power rule to solveequations like

2x -1/3 = 6 ?

x1 = x

raise x -1/3 to the reciprocal power

x 1 3 3 1x

but first . . . .


simplify

isolate the variable with the exponent

Using the Power of Power Rule

Power of Power Property (am)n = am•n

2x -1/3 = 62 2

x -1/3 = 3

multiply both sidesby the exponent’s reciprocal

(x -1/3)-3 = 3-3

x = 1/33 = 1/27

Check your answer

but first . . . .


Model Problem

Solve for x3/2 + 1 = 9

isolate variable with acoefficient of 1

raise both members of equation by reciprocal power

simplify

x3/2 + 1 = 9-1 -1

x3/2 = 8

(x3/2)2/3 = 82/3

x = 82/3

x ( 83 )2 4


Rational Exponents containing variables

Solve and check: 5x + 1 = 54

for b 0 and b 1, bx = by x = y

because the base on both sides of thisequation is 5, we can write the following:

x + 1 = 4

x = 3

53 + 1 = 54check:

54 = 54



Solve and check: 2x – 1 = 82

check:26 = 82

change the right side to base 2

2x – 1 = (23)2

23 = 8

simplify 2x – 1 = 26

equate exponents x – 1 = 6

solve x = 7

64 = 64

27 – 1 = 82

bx = by x = y



Solve and check:

check:

change both sides to base 2 (2-2)x = (23)1 – x

1/4 = 2-2

simplify 2-2x = 23 – 3x

equate exponents -2x = 3 – 3x

solve x = 3

8 = 23

1

4

x

81 x

1

4

3

81 3

1

648 2

1

82 1

64



Solve and check: 9x + 1 = 27x

check:93 = 272

change both sides to base 3 (32)x + 1 = (33)x

32 = 9

simplify 32x + 2 = 33x

equate exponents 2x + 2 = 3x

solve x = 2

729 = 729

33 = 27

92 + 1 = 272

Aim: Exponential Equations Course: Alg. 2 & Trig. Aim: How do we solve equations with fractions, negative numbers, or variables in the exponents? Do Now:

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