1. 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?

2. square root of both sides
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
T  2.8 hours

3. How do we use this power rule to solve
Power of Power Rule
Power of Power Property
(am)n = am•n
(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
but first

4. Using the Power of Power Rule
Power of Power Property
(am)n = am•n
but first
isolate the variable
with the exponent
2x -1/3 = 6
x -1/3 = 3
multiply both sides
by the exponent’s
reciprocal
(x -1/3)-3 = 3-3
simplify
x = 1/33 = 1/27
Check your answer

5. Model Problem Solve for x3/2 + 1 = 9 x3/2 + 1 = 9 -1 -1 x3/2 = 8
isolate variable with a
coefficient of 1
x3/ = 8
raise both members of
equation by reciprocal power
(x3/2)2/3 = 82/3
x = 82/3
simplify

6. 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 this
equation is 5, we can write the following:
x + 1 = 4
x = 3
check:
= 54
54 = 54

7. Rational Exponents containing variables
Solve and check: 2x – 1 = 82
bx = by  x = y
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
check:
27 – 1 = 82
26 = 82
64 = 64

8. Rational Exponents containing variables
Solve and check:
change both
sides to base 2
(2-2)x = (23)1 – x
1/4 = 2-2
8 = 23
simplify
2-2x = 23 – 3x
equate exponents
-2x = 3 – 3x
solve
x = 3
check:

9. Rational Exponents containing variables
Solve and check: 9x + 1 = 27x
change both
sides to base 3
(32)x + 1 = (33)x
32 = 9
33 = 27
simplify
32x + 2 = 33x
equate exponents
2x + 2 = 3x
solve
x = 2
check:
= 272
93 = 272
729 = 729