1 topic 1.3.1 exponent laws. 2 topic 1.3.1 exponent laws california standard: 2.0 students...

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Topic 1.3.1Topic 1.3.1

Exponent LawsExponent Laws

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Topic1.3.1


California Standard: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

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Topic1.3.1


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.

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Topic1.3.1

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.

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Topic1.3.1

Example 1

Solution follows…


a) 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

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Topic1.3.1

Guided Practice

Solution follows…


Expand each expression and evaluate.

1. 23

3. 52 × 32

2. 32

4. 24y3

24y3 = 2 × 2 × 2 × 2 × y × y × y = 16y3

23 = 2 × 2 × 2 = 8

5. Find the area, A, of the square shown.Write your answer as a power of s.

6. If the sides of the square are 7 inches long, what is the area?

7. Find the volume of a cube if the edges are 2 feet long.(Volume V = e3, where e is the edge length.)

s

s

A = s2

A = 49 inches2

V = 8 ft3

32 = 3 × 3 = 9

52 × 32 = 5 × 5 × 3 × 3 = 225

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ma × mb = ma + b

Topic1.3.1

There are Lots of Rules of Exponents


1) 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.

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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

ma ÷ mb = ma – b

Topic1.3.1


2) In a similar way, to divide powers, you subtract the exponents.

3) When you raise a power to a power, you multiply the exponents — for example, (m3)2 = m3 × m3 = m6. (ma)b = mab

mb

mb

a

=a

a

ma × mb = ma + b 1)

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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.

(mb)a = maba

ma ÷ mb = ma – b

Topic1.3.1


(ma)b = mab

m0 = 15) Using rule 1 above:

ma × m0 = ma + 0 = ma. So m0 equals 1.

ma × mb = ma + b 1) 2)

3) 4)

1ma(ma)–1 = m–a =

7) And taking a root can be written using a fractional power. a

n = a

1n

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Topic1.3.1


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.

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Topic1.3.1

Independent Practice

Solution follows…


In Exercises 1–6, write each expression using exponents.

1. 2 × 2 × 2 × 2

2. a × a × a × 4

3. 2 × k × 2 × 2 × k

4. 4 × 3 × 3 × 4 × p × 3 × 3 × p × 4

5. a × b × a × b

6. 5 × l × 3 × 5 × 5 × l

= 24

= 4a3

= 23k2

= a2b2

= 3 • 53 • l2

= 34 • 43 • p2

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Topic1.3.1


Solution follows…


7. Show that = k2. k6

k4

k6 k • k • k • k • k • k

k4 k • k • k • k= = k • k = k2

1 1 1 1

1 1 1 1

Simplify the expressions in Exercises 8–16 using rules of exponents.

8. 170 9. 2–3 10. 22 • 23

11. 12. (23)2 • 22 13.

14. 15. (x4 ÷ x2) • x3 16. (x2)3 ÷ x4

118

32

936

34

23 • 34

37

(32)2

33

256

3

827

x2x5

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Topic1.3.1


Solution follows…


Simplify the expressions in Exercises 17–25 using rules of exponents.

x3 • x5

(ax)2

(x3)–3

x–4• x517. 18. 19. (2x–2)3 • 4x2

20. 3x0y–2 21. (3x)0xy–2 22. 5x–1 × 6(xy)0

23. 24. 25.(4x)2y

2x

(2x3)2y

y–2

(32x5y3)–2

x4y–6

x6

a21

32

x4

3

y2

x

y2

30

x

8xy 4x6y3 1

81x14

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Topic1.3.1


Solution follows…


26. An average baseball has a radius, r, of 1.45 inches.Find the volume, V, of a baseball in cubic inches.

12.77 inches3

(V = r3)43

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?

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 = at212

E = mv212

50 joules

62.5 m/s

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Topic1.3.1

Round UpRound 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.

an = a

1n

1 topic 1.3.1 exponent laws. 2 topic 1.3.1 exponent laws california standard: 2.0 students...

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