1. Exponent Laws
Topic 1.3.1

2. Exponent Laws 1.3.1 Topic California Standard: What it means for you:
2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.
What it means for you:
You’ll learn about the rules of exponents.
Key words:
exponent
base
power
product
quotient

3. Topic
1.3.1
Exponent Laws
Exponents have a whole set of rules to make sure that all mathematicians deal with them in the same way.
There are lots of rules written out in this Topic, so take care.

4. Exponent Laws 1.3.1 Topic Powers are Repeated Multiplications
A power is a multiplication in which all the factors are the same.
For example, m2 = m × m and m3 = m × m × m are both powers of m.
In this kind of expression, “m” is called the base and the “2” or “3” is called the exponent.

5. Topic
1.3.1
Exponent Laws
Example 1
Find the volume of the cube shown. Write your answer as a power of e.
b) If the edges of the cube are 4 cm long, what is the volume?
Solution
a) V = e × e × e = e3
b) V = e3 = (4 cm)3
= 43 cm3
= 64 cm3
Solution follows…

6. Exponent Laws 1.3.1 Topic Guided Practice
Expand each expression and evaluate.
1. 23
3. 52 × 32
2. 32
4. 24y3
23 = 2 × 2 × 2 = 8
32 = 3 × 3 = 9
52 × 32 = 5 × 5 × 3 × 3 = 225
24y3 = 2 × 2 × 2 × 2 × y × y × y = 16y3
5. Find the area, A, of the square shown. Write your answer as a power of s. s
A = s2
6. If the sides of the square are 7 inches long, what is the area?
A = 49 inches2
7. Find the volume of a cube if the edges are 2 feet long. (Volume V = e3, where e is the edge length.)
V = 8 ft3
Solution follows…

7. Exponent Laws 1.3.1 Topic There are Lots of Rules of Exponents
If you multiply m2 by m3, you get m5, since: m2 × m3 = (m × m) × (m × m × m) = m × m × m × m × m = m5
The exponent of the product is the same as the exponents of the factors added together.
This result always holds — to multiply powers with the same base, you simply add the exponents.
ma × mb = ma + b

8. Exponent Laws 1.3.1 m = b Topic 1) ma × mb = ma + b
2) In a similar way, to divide powers, you subtract the exponents.
ma ÷ mb = ma – b
3) When you raise a power to a power, you multiply the exponents — for example, (m3)2 = m3 × m3 = m6.
(ma)b = mab
4) Raising a product or quotient to a power is the same as raising each of its elements to that power. For example: (mb)3 = mb × mb × mb = (m × b) × (m × b) × (m × b) = m × m × m × b × b × b = m3b3
(mb)a = maba m b a =

9. Exponent Laws 1.3.1 Topic 1) ma × mb = ma + b 2) ma ÷ mb = ma – b 3)
(ma)b = mab 4)
(mb)a = maba
5) Using rule 1 above: ma × m0 = ma + 0 = ma. So m0 equals 1.
m0 = 1
6) It’s also possible to make sense of a negative exponent. ma × m–a = ma – a = m0 = 1 (using rules 1 and 5 above)
So the reciprocal of ma is m–a. 1 m a
(ma)–1 = m–a =
7) And taking a root can be written using a fractional power. a n Ö = 1

10. Topic
1.3.1
Exponent Laws
These rules always work, unless the base is 0.
The exponents and the bases can be positive, negative, whole numbers, or fractions. The only exception is you cannot raise zero to a negative exponent — zero does not have a reciprocal.

11. Exponent Laws 1.3.1 Topic Independent Practice
In Exercises 1–6, write each expression using exponents.
1. 2 × 2 × 2 × 2
= 24
2. a × a × a × 4
= 4a3
3. 2 × k × 2 × 2 × k
= 23k2
4. 4 × 3 × 3 × 4 × p × 3 × 3 × p × 4
= 34 • 43 • p2
5. a × b × a × b
= a2b2
6. 5 × l × 3 × 5 × 5 × l
= 3 • 53 • l2
Solution follows…

12. Exponent Laws 1.3.1 Topic Independent Practice 7. Show that = k2. k6
k k • k • k • k • k • k
k k • k • k • k =
= k • k = k2 1
7. Show that = k2. k6 k4
Simplify the expressions in Exercises 8–16 using rules of exponents. 1 8
8. 170 1
9. 2–3
• 23 32 36 34
23 • 34 37 8 27
11. 9
12. (23)2 • 22
256
13.
(32)2 33
14.
15. (x4 ÷ x2) • x3
16. (x2)3 ÷ x4 3 x5 x2
Solution follows…

13. Exponent Laws 1.3.1 Topic Independent Practice
Simplify the expressions in Exercises 17–25 using rules of exponents.
x3 • x5
(ax)2
(x3)–3
x–4
• x5 x6 a2 32 x4
17.
18. 1
19. (2x–2)3 • 4x2 3 y2 x y2 30 x
20. 3x0y–2
21. (3x)0xy–2
22. 5x–1 × 6(xy)0
(4x)2y 2x
(2x3)2y
y–2
(32x5y3)–2
x4y–6 1
81x14
23.
24.
25.
8xy
4x6y3
Solution follows…

14. Exponent Laws 1.3.1 Topic Independent Practice
26. An average baseball has a radius, r, of 1.45 inches. Find the volume, V, of a baseball in cubic inches.
(V = pr3) 4 3
12.77 inches3
E = mv2 1 2
27. The kinetic energy of a ball (in joules) is given by
where m is the ball’s mass (in kilograms) and v is its velocity (in meters per second). If a ball weighs 1 kilogram and is traveling at 10 meters per second, what is its kinetic energy in joules?
50 joules
28. The speed of a ball (in meters per second) accelerating from rest
is given by , where a is its acceleration (in meters per
second squared) and t is its time of flight (in seconds). Calculate the speed of a ball in meters per second after 5 seconds of flight if it is accelerating at 5 meters per second squared.
v = at2 1 2
62.5 m/s
Solution follows…

15. Exponent Laws 1.3.1 Topic Round Up
That’s a lot of rules, but don’t worry — you’ll get plenty of practice using them later in the program.
Exponents often turn up when you’re dealing with area and volume.
The next Topic will deal just with square roots, which is a special case of Rule 7. a n Ö = 1