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Lesson 2.5.1Lesson 2.5.1

Powers of IntegersPowers of Integers

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2.5.1Powers of IntegersPowers of Integers

California Standards:Number Sense 1.2Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers.

Algebra and Functions 2.1Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents.

What it means for you:You’ll learn how to write repeated multiplications in a shorter form.

Key words:• power• base• exponent• factor

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2.5.1Powers of IntegersPowers of Integers

A power is just the product that you get when you repeatedly multiply a number by itself, like 2 • 2, or 3 • 3 • 3.

Repeated multiplication expressions can be very long.

So there’s a special system you can use for writing out powers in a shorter way — that’s what this Lesson is about.

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A Power is a Repeated Multiplication

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2.5.1Powers of IntegersPowers of Integers

A power is a product that results from repeatedly multiplying a number by itself.

For example:

2 • 2 = 4, or “two to the second power.”

2 • 2 • 2 = 8, or “two to the third power.”

2 • 2 • 2 • 2 = 16, or “two to the fourth power.”

So 4, 8, and 16 are all powers of 2.

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You Can Write a Power as a Base and an Exponent

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2.5.1Powers of IntegersPowers of Integers

If every time you used a repeated multiplication you wrote it out in full, it would make your work very complicated. So there’s a shorter way to write them.

For example:

2 • 2 • 2 • 2 = 24

2 is the base — it’s the number that’s being multiplied

This is the exponent — it tells you how many times the base number is a factor in the multiplication expression.

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The expression 10 • 10 can be written in this form too — the base is 10 and since 10 occurs twice, the exponent is 2.

102

Base

Exponent

You can rewrite any repeated multiplication in this form. So any number, x, to the nth power can be written as:

xn

Base

Exponent

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Example 1

Solution follows…

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2.5.1Powers of IntegersPowers of Integers

Write the expression: 3 • 3 • 3 • 3 • 3 in base and exponent form.

Solution

The number that is being multiplied is 3. So the base is 3.

So 3 • 3 • 3 • 3 • 3 = 35.

3 occurs as a factor five times in the multiplication expression. So the exponent is 5.

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If a number has an exponent of 1 then it occurs only once in the expanded multiplication expression.

So any number to the power 1 is just the number itself.

For example: 51 = 5,

1371 = 137,

x1 = x.

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Guided Practice

Solution follows…

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2.5.1Powers of IntegersPowers of Integers

Write each of the expressions in Exercises 1–8 as a power in base and exponent form.

1. 8 • 8 2. 2 • 2 • 2

3. 7 • 7 • 7 • 7 • 7 4. 5

5. 9 • 9 • 9 • 9 6. 4 • 4 • 4 • 4 • 4 • 4 • 4

7. –5 • –5 8. –8 • –8 • –8 • –8 • –8

82 23

75 51

94 47

(–5)2 (–8)5

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Evaluate a Power By Doing the Multiplication

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2.5.1Powers of IntegersPowers of Integers

Evaluating a power means working out its value.

Just write it out in its expanded form — then treat it as any other multiplication calculation.

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Example 2

Solution follows…

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2.5.1Powers of IntegersPowers of Integers

Evaluate 54.

Solution

54 means “four copies of the number five multiplied together.”

54 = 5 • 5 • 5 • 5

54 = 625

Write the expression in expanded form

Then do the multiplication

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Example 3

Solution follows…

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2.5.1Powers of IntegersPowers of Integers

Evaluate (–2)2.

Solution

(–2)2 means “two copies of negative two multiplied together.”

(–2)2 = –2 • –2

(–2)2 = 4

Write the expression in expanded form

Then do the multiplication

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Guided Practice

Solution follows…

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2.5.1Powers of IntegersPowers of Integers

Evaluate the exponential expressions in Exercises 9–16.

9. 102 10. 53

11. 71 12. 36

13. 471 14. (–15)1

15. (–3)2 16. (–4)3

10 • 10 = 100 5 • 5 • 5 = 125

7 3 • 3 • 3 • 3 • 3 • 3 = 729

47 –15

–3 • –3 = 9 –4 • –4 • –4 = –64

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Independent Practice

Solution follows…

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2.5.1Powers of IntegersPowers of Integers

Write each of the expressions in Exercises 1–6 in base and exponent form.

1. 4 • 4 • 4 2. 9 • 9

3. 8 4. 5 • 5 • 5 • 5 • 5 • 5

5. –4 • –4 • –4 6. –3 • –3 • –3 • –3

43 92

81 56

(–4)3 (–3)4

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Independent Practice

Solution follows…

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2.5.1Powers of IntegersPowers of Integers

7. Kiera and 11 of her friends are handing out fliers for a school fund-raiser. Each person hands out fliers to 12 people. How many people receive a flier?

12 • 12 = 122 = 144

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Independent Practice

Solution follows…

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2.5.1Powers of IntegersPowers of Integers

Evaluate the exponential expressions in Exercises 8–13.

8. 152 9. 43

10. 81 11. 18

12. (–5)3 13. (–5)4

225 64

8 1

–125 625

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Independent Practice

Solution follows…

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2.5.1Powers of IntegersPowers of Integers

14. A single yeast cell is placed on a nutrient medium. This cell will divide into two cells after one hour. These two cells will then divide to form four cells after another hour. The process continues indefinitely.

a) How many yeast cells will be present after 1 hour, 2 hours, and 6 hours?

b) Write exponential expressions with two as the base to describe the number of yeast cells that will be present after 1 hour, 2 hours, and 6 hours.

c) How many hours will it take for the yeast population to reach 256?

1 hour = 2 cells, 2 hours = 4 cells, 6 hours = 64 cells

1 hour = 21 cells, 2 hours = 22 cells, 6 hours = 26 cells

8 hours

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Round UpRound Up

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2.5.1Powers of IntegersPowers of Integers

If you need to use a repeated multiplication, it’s useful to have a shorter way of writing it.

That’s why bases and exponents come in really handy when you’re writing out powers of numbers.

You’ll see lots of powers used in expressions, equations, and formulas. For example, the formula for the area of a circle is r2 where r is the radius.So it’s important you know what they mean.