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?
Aim: Exponential Equations Course: Alg. 2 & Trig.
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
Aim: Exponential Equations Course: Alg. 2 & Trig.
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 . . . .
Aim: Exponential Equations Course: Alg. 2 & Trig.
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 . . . .
Aim: Exponential Equations Course: Alg. 2 & Trig.
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
Aim: Exponential Equations Course: Alg. 2 & Trig.
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
Aim: Exponential Equations Course: Alg. 2 & Trig.
Rational Exponents containing variables
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
Aim: Exponential Equations Course: Alg. 2 & Trig.
Rational Exponents containing variables
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
Aim: Exponential Equations Course: Alg. 2 & Trig.
Rational Exponents containing variables
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
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