Transcript Solving Equations with Exponents

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
substitute D = 12
123 = 216T2
simplify and solve
1728 = 216T2
8 = T2
square root of both sides
8  T2
T  2.8 hours
Aim: Exponential Equations
Course: Alg. 2 & Trig.
Power of Power Rule
(am)n = am•n
Power of Power Property
(x2)1/2 = x1 = x
(x -1/2)-2 = x1 = x
How do we use this power rule to solve
equations like
2x -1/3 = 6 ?
raise x -1/3 to the reciprocal power
x 
1 3 3 1
x
but first . . . .
Aim: Exponential Equations
Course: Alg. 2 & Trig.
Using the Power of Power Rule
Power of Power Property
(am)n = am•n
but first . . . .
2x -1/3 = 6
2
2
isolate the variable
with the exponent
x -1/3 = 3
(x -1/3)-3 = 3-3
x = 1/33 = 1/27
multiply both sides
by the exponent’s
reciprocal
simplify
Check your answer
Aim: Exponential Equations
Course: Alg. 2 & Trig.
Model Problem
Solve for x3/2 + 1 = 9
x3/2 + 1 = 9
-1 -1
x3/2
(x3/2)2/3
x
=8
=
82/3
isolate variable with a
coefficient of 1
raise both members of
equation by reciprocal power
= 82/3
x  (3 8)2  4
simplify
Aim: Exponential Equations
Course: Alg. 2 & Trig.
Rational Exponents containing variables
5x + 1 = 5 4
Solve and check:
for b  0 and b  1, bx = by  x = y
because the base on both sides of this
equation is 5, we can write the following:
x+1=4
x=3
check:
53 + 1 = 5 4
54 = 5 4
Aim: Exponential Equations
Course: Alg. 2 & Trig.
Rational Exponents containing variables
2x – 1 = 8 2
Solve and check:
bx = b y  x = y
change the right
side to base 2
23 = 8
simplify
2x – 1 = (23)2
2x – 1 = 26
equate exponents
x –1=6
solve
x
check:
=7
27 – 1 = 8 2
26 = 8 2
64 = 64
Aim: Exponential Equations
Course: Alg. 2 & Trig.
Rational Exponents containing variables
x
Solve and check:
1 
1 x
8
4 
change both
sides to base 2
1/4 = 2-2 8 = 23
(2-2)x = (23)1 – x
2-2x = 23 – 3x
simplify
equate exponents
solve
-2x
x
3
check: 1   81 3
4 
Aim: Exponential Equations
= 3 – 3x
=3
1
1
1
2
8  2 
64
8
64
Course: Alg. 2 & Trig.
Rational Exponents containing variables
9x + 1 = 27x
Solve and check:
change both
sides to base 3
32 = 9 33 = 27
(32)x + 1 = (33)x
32x + 2 = 33x
simplify
equate exponents
2x + 2 = 3x
solve
x
check:
=2
92 + 1 = 272
93 = 272
729 = 729
Aim: Exponential Equations
Course: Alg. 2 & Trig.