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Lesson 9-3 Warm-Up

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“Finding and Estimating Square Roots” (9-3)

(9-2)What is the “square root” of a number?

What are the parts of “square root”?

square root: The square root, a, of a number, b, is the number that equals b when squared.

a is the square root of b in a2 = b

The diagram below shows the relationship between squares (a number multiplied by itself) and square roots (the opposite of squaring or the number that needs to be multiplied by itself to equal a given number)

Notice that every positive number has two square roots (since a negative times a negative equals a positive)

A radical symbol indicate a square root. The number inside the radical sign is called the radicand. You can indicate that a number can be positive or negative with a , which means “plus or minus”. The square root of a negative number is undefined (impossible) meaning you can’t have a negative number inside a radical sign. However, the negative can be outside the radical. This is called a negative square root.(Example: - 16 means “the negative square root of 16.”)

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“Finding and Estimating Square Roots” (9-3)

(9-2)Examples:

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a. 25

Simplify each expression.

c. – 64

d. –49

b. ± 925

positive square root= 5

35

35

35

The square roots are and – .= ±

negative square root= –8

For real numbers, the square root of a negative number is undefined.

is undefined

e. 116

14

14

The square root is .=

Finding and Estimating Square RootsLESSON 9-3

Additional Examples

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“Finding and Estimating Square Roots” (9-3)

(9-2)How can you tell if the square root of a number is a rational or irrational number.

The square root of a number is rational if its decimal form terminates (ends) or repeats.

Examples:

The square root of a number is irrational if its decimal form continues without repeating.

Examples:

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a. ± 144 = ± 12

Tell whether each expression is rational or irrational.

c. – 6.25 = –2.5

e. 7 = 2.64575131 . . .

b. = – 0.44721359…– 1 5

d. = 0.319

rational

irrational

rational

rational

irrational

Finding and Estimating Square RootsLESSON 9-3

Additional Examples

ALGEBRA 1

“Finding and Estimating Square Roots” (9-3)

(9-2)What is a perfect square?

How can you estimate the square root of a number?.

perfect square: the square of an integer (in other words, an integer times itself)

Examples:

You can estimate the square root of a number by figuring out what perfect squares its between.

Examples: Between what two consecutive integers is 14.52 .

14.52 is between 3 and 4.

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Between what two consecutive integers is 28.34 ?

28.34 is between 5 and 6.

28.34 is between the two consecutive perfect squares 25 and 36.

25 < 28.34 < 36

The square roots of 25 and 36 are 5 and 6, respectively.

28.345 < < 6

Finding and Estimating Square RootsLESSON 9-3

Additional Examples

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Suppose a rectangular field has a length three times its

width x. The formula d = x2 + (3x)2 gives the distance of the

diagonal of a rectangle. Find the length of the diagonal if x = 8 ft.

d = x2 + (3x)2

d = (8)2 + (3 • 8)2 Substitute 8 for x.

The diagonal is about 25.3 ft long.

Use a calculator. Round to the nearest tenth.d 25.3

Simplify using PEMDAS (2. Exponents)d = 64 + 576

d = 640

Finding and Estimating Square RootsLESSON 9-3

Additional Examples

d = (8)2 + (24)2 Simplify using PEMDAS (1. Parenthesis)

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425

35

14 ±25

irrational undefined rational

–8 and –7

about 40.25 mph

1. Simplify each expression.

a. 196        b.  ±

2. Tell whether each expression is rational, irrational, or undefined.

a. ±       b.  –25       c. – 2.25

3. Between what two consecutive integers is – 54?

4. The formula s = 13.5d estimates the speed s in miles per hour that a car was traveling, when it applied its brakes and left a skid mark d feet long on a wet road. Estimate the speed of a car that left a 120-foot- long skid mark.

Finding and Estimating Square RootsLESSON 9-3

Lesson Quiz