4. Simplify the following expressions completely.
8 3
(6x5 )(5x6 )
a
2 4
(a.)
(b.) (a b )
−7
3
(x )
2b
4A:
(6x5 )(5x6 )
(x−7 )3
4B: (a2 b4 )
4C:
a8
2b
30x11
(x−7 )3
—Prop 1−→
3
p
3
27p12 q 15
—Prop 2−→
—Prop 4−→
(a2 b4 )
(c.)
—Prop 2−→
a24
23 b3
(27p12 q 15 )1/3
30x11
x−21
—Prop 1−→
—Prop 2−→
p
3
27p12 q 15
—Prop 3−→
30x32
a26 b4
—Prop 3−→
8b3
a26 b1
8
271/3 p4 q 5 = 3p4 q 5
5. Simplify the following as an exponential term with a base of 3. For example, (35 )4 = 320 .
√ 4
√
1
(a.)
(b.) 27
(c.) 5 9
81
1
1
= 4 = 3−4
81
3
√
27 = (27)1/2 = (33 )1/2 = 33/2
√
4
5
9 = (9)4/5 = (32 )4/5 = 38/5
6. Solve for x for the following equations. Hint: Simplify each side of the equation so that you can use property #6.
√
2
(a.) 43x+3 = 45x−8
(b.) 36x +1 = 9
(c.) 72x+2 = 343 · 7
2
(a.) 3x + 3 = 5x − 8
(b.) 36x +1 = 32
(c.) 72x+2 = 73 · 71/2
2
(a.) 11 = 2x
(b.) 6x + 1 = 2
(c.) 72x+2 = 73+1/2
2
(a.) 11/2 = x
(b.) 6x = 1
(c.) 2x + 2 = 3.5
1
2
(b.) x =
(c.) 2x = 1.5
6r
1
1.5
3
(b.) x = ±
(c.) x =
=
6
2
4
Exponential Functions.
Exponential functions are of the form y = ax where a is a positive base and x is a variable. There are two types of
exponential graphs:
For example, y = 3x would look like the graph on the left, while y = 0.7x would look like the graph on the right.
Comparng and Contrasting the Two Types of Exponential Graphs.
Please fill in the following table regarding the two graphs above.
Domain
Range
x-intercept
y-intercept
Increasing or
Decreasing
Type I: a > 1:
(−∞, ∞)
(0, ∞)
none
(0, 1)
increasing
Type II: a < 1:
(−∞, ∞)
(0, ∞)
none
(0, 1)
decreasing
Note: For both graphs, we have a horizontal asymptote at the x-axis, and the entire graph lies above this line.
Examples:
7. Use what you know about graph reflections and translations to give a rough sketch of the following functions. Make
sure to draw where the horizontal asymptote ends up.
(a) y = 3x−4 + 5
(b) y =
2−x
1
−4
5
(c) y = −4−x−3 − 1
Domain: (−∞, ∞)
Domain: (−∞, ∞)
Domain: (−∞, ∞)
Range: (5, ∞)
Range: (−4, ∞)
Range: (−∞, −1)