1-5 Roots and Irrational Numbers

Warm UpWarm Up

Lesson PresentationLesson Presentation

California Standards

PreviewPreview

1-5 Roots and Irrational Numbers

Warm UpSimplify each expression.

1. 62 36 2. 112 121

3. (–9)(–9) 81 4. 2536

Write each fraction as a decimal.

5. 25

596.

7. 5 38

8. –1 56

0.4

5.375

0.5

–1.83

1-5 Roots and Irrational Numbers

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.

California Standards

1-5 Roots and Irrational Numbers

square root terminating decimalprincipal square root repeating decimalperfect square irrational numberscube root natural numberswhole numbersintegersrational numbers

Vocabulary

1-5 Roots and Irrational Numbers

4 4 = 42 = 16 = 4 Positive squareroot of 16

(–4)(–4) = (–4)2 = 16 = –4 Negative squareroot of 16

–

A number that is multiplied by itself to form a product is a square root of that product. The radical symbol is used to represent square roots. For nonnegative numbers, the operations of squaring and finding a square root are inverse operations. In other words, for x ≥ 0,

Positive real numbers have two square roots.

1-5 Roots and Irrational Numbers

A perfect square is a number whose positive square root is a whole number. Some examples of perfect squares are shown in the table.

0

02

1

12

1004

22

9

32

16

42

25

52

36

62

49

72

64

82

81

92 102

The principal square root of a number is the positive square root and is represented by . A negative square root is represented by – . The symbol is used to represent both square roots.

1-5 Roots and Irrational Numbers

The small number to the left of the root is the index. In a square root, the index is understood to be 2. In other words, is the same as .

Writing Math

1-5 Roots and Irrational Numbers

A number that is raised to the third power to form a product is a cube root of that product. The symbol indicates a cube root. Since 23 = 8, = 2. Similarly, the symbol indicates a fourth root: 2 = 16, so = 2.

1-5 Roots and Irrational Numbers

Additional Example 1: Finding Roots

Find each root.

Think: What number squared equals 81?

Think: What number squared equals 25?

1-5 Roots and Irrational Numbers

Find the root.

Think: What number cubed equals –216?

Additional Example 1: Finding Roots

= –6 (–6)(–6)(–6) = 36(–6) = –216

C.

1-5 Roots and Irrational Numbers

Find each root.

Check It Out! Example 1

Think: What number squared equals 4?

Think: What number squared equals 25?

a.

b.

1-5 Roots and Irrational Numbers

Find the root.

Check It Out! Example 1

Think: What number to the fourth power equals 81?

c.

1-5 Roots and Irrational Numbers

Additional Example 2: Finding Roots of Fractions

Find the root.

Think: What number squared

equals

A.

1-5 Roots and Irrational Numbers

Additional Example 2: Finding Roots of Fractions

Find the root.

Think: What number cubed equals

B.

1-5 Roots and Irrational Numbers

Additional Example 2: Finding Roots of Fractions

Find the root.

Think: What number squared

equals

C.

1-5 Roots and Irrational Numbers

Find the root.

Check It Out! Example 2

Think: What number squared equals

a.

1-5 Roots and Irrational Numbers

Find the root.

Check It Out! Example 2

Think: What number cubed equals

b.

1-5 Roots and Irrational Numbers

Find the root.

Check It Out! Example 2c

Think: What number squared equals

c.

1-5 Roots and Irrational Numbers

Square roots of numbers that are not perfect squares, such as 15, are not whole numbers. A calculator can approximate the value of as 3.872983346... Without a calculator, you can use square roots of perfect squares to help estimate the square roots of other numbers.

1-5 Roots and Irrational Numbers

Additional Example 3: Art Application

As part of her art project, Shonda will need to make a paper square covered in glitter. Her tube of glitter covers 13 in². Estimate to the nearest tenth the side length of a square with an area of 13 in².

Since the area of the square is 13 in², then each side of the square is in. 13 is not a perfect square, so find two consecutive perfect squares that is between: 9 and 16. is between and , or 3 and 4. Refine the estimate.

1-5 Roots and Irrational Numbers

Additional Example 3 Continued

3.5 3.52 = 12.25 too low

3.6 3.62 = 12.96 too low

3.65 3.652 = 13.32 too high

The side length of the paper square is

Since 3.6 is too low and 3.65 is too high, is between 3.6 and 3.65. Round to the nearest tenth.

1-5 Roots and Irrational Numbers

The symbol ≈ means “is approximately equal to.”

Writing Math

1-5 Roots and Irrational Numbers

What if…? Nancy decides to buy more wildflower seeds and now has enough to cover 26 ft2. Estimate to the nearest tenth the side length of a square garden with an area of 26 ft2.

Check It Out! Example 3

Since the area of the square is 26 ft², then each side of the square is ft. 26 is not a perfect square, so find two consecutive perfect squares that is between: 25 and 36. is between and , or 5 and 6. Refine the estimate.

1-5 Roots and Irrational Numbers

Check It Out! Example 3 Continued

5.0 5.02 = 25 too low

5.1 5.12 = 26.01 too high

Since 5.0 is too low and 5.1 is too high, is between 5.0 and 5.1. Rounded to the nearest tenth, 5.1. The side length of the square garden is 5.1 ft.

1-5 Roots and Irrational Numbers

Real numbers can be classified according to their characteristics.

Natural numbers are the counting numbers: 1, 2, 3, …

Whole numbers are the natural numbers and zero: 0, 1, 2, 3, …

Integers are the whole numbers and their opposites: –3, –2, –1, 0, 1, 2, 3, …

1-5 Roots and Irrational Numbers

Rational numbers are numbers that can be expressed in the form , where a and b are both integers and b ≠ 0. When expressed as a decimal, a rational number is either a terminating decimal or a repeating decimal.

• A terminating decimal has a finite number of digits after the decimal point (for example, 1.25, 2.75, and 4.0).

• A repeating decimal has a block of one or more digits after the decimal point that repeat continuously (where all digits are not zeros).

1-5 Roots and Irrational Numbers

Irrational numbers are all numbers that are not rational. They cannot be expressed in the form where a and b are both integers and b ≠ 0. They are neither terminating decimals nor repeating decimals. For example:

0.10100100010000100000…

After the decimal point, this number contains 1 followed by one 0, and then 1 followed by two 0’s, and then 1 followed by three 0’s, and so on.

This decimal neither terminates nor repeats, so it is an irrational number.

1-5 Roots and Irrational Numbers

If a whole number is not a perfect square, then its square root is irrational. For example, 2 is not a perfect square and is irrational.

1-5 Roots and Irrational Numbers

The real numbers are made up of all rational and irrational numbers.

1-5 Roots and Irrational Numbers

Note the symbols for the sets of numbers.R: real numbersQ: rational numbersZ: integersW: whole numbersN: natural numbers

Reading Math

1-5 Roots and Irrational Numbers

Additional Example 4: Classifying Real Numbers

Write all classifications that apply to each real number.

A.

–32 = –

32 1

rational number, integer, terminating decimal

B.

irrational

–32

–32 can be written in the form .

14 is not a perfect square, so is irrational.

–32 can be written as a terminating decimal.

–32 = –32.0

1-5 Roots and Irrational Numbers

Write all classifications that apply to each real number.

a. 7

rational number, repeating decimal

Check It Out! Example 4

67 9 = 7.444… = 7.4

7 can be written in the form .49

can be written as a repeating decimal.

b. –12 –12 can be written in the form .

–12 can be written as a terminating decimal.

rational number, terminating decimal, integer

1-5 Roots and Irrational Numbers

Write all classifications that apply to each real number.

Check It Out! Example 4

irrational

100 is a perfect square, so is rational.

10 is not a perfect square, so is irrational.

10 can be written in the form and as a terminating decimal.

natural, rational, terminating decimal, whole, integer

1-5 Roots and Irrational Numbers

Find each square root.

1. 2. 3. 4.3

5. The area of a square piece of cloth is 68 in2. Estimate to the nearest tenth the side length of the cloth. 8.2 in.

Lesson Quiz

Write all classifications that apply to each real number.

6. –3.89 7.rational, repeating decimal

irrational

15