Evaluating Algebraic Expressions

4-6 Squares and Square Roots

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Evaluating Algebraic Expressions

4-6 Squares and Square Roots

Warm UpSimplify.

25 64

144 225

400

1. 52 2. 82

3. 122 4. 152

5. 202

Evaluating Algebraic Expressions

4-6 Squares and Square Roots

M8N1. Students will understand different representations of numbers including

square roots, exponents, and scientific notation.

GPS

Evaluating Algebraic Expressions

4-6 Squares and Square Roots

a. Find square roots of perfect squares.

b. Recognize the (positive) square root of a number as a length of a side of a square with a given area.

c. Recognize square roots as points and as lengths on a number line.

GPS

Evaluating Algebraic Expressions

4-6 Squares and Square Roots

GPS• d. Understand that the square root of 0 is 0

and that every positive number has two square roots that are opposite in sign.

• e. Recognize and use the radical symbol to denote the positive square root of a

Evaluating Algebraic Expressions

4-6 Squares and Square Roots

Vocabularysquare root

principal square root

perfect square

radical

Evaluating Algebraic Expressions

4-6 Squares and Square Roots

The square root of a number is one of the two equal factors of that number. Squaring a nonnegative number and finding the square root of that number are inverse operations.

Because the area of a square can be expressed using an exponent of 2, a number with an exponent of 2 is said to be squared. You read 32 as “three squared.”

3

3

Area = 32

Evaluating Algebraic Expressions

4-6 Squares and Square Roots

Positive real numbers have two square roots, one positive and one negative. The positive square root, or principal square root, is represented by . The negative square root is represented by – .

Evaluating Algebraic Expressions

4-6 Squares and Square Roots

A perfect square is a number whose square roots are integers. Some examples of perfect squares are shown in the table.

Evaluating Algebraic Expressions

4-6 Squares and Square Roots

You can write the square roots of 16 as ±4, which is read as “plus or minus four.”

Writing Math

Evaluating Algebraic Expressions

4-6 Squares and Square RootsAdditional Example: 1 Finding the Positive and Negative

Square Roots of a NumberFind the two square roots of each number.

7 is a square root, since 7 • 7 = 49.

–7 is also a square root, since –7 • (–7) = 49.

10 is a square root, since 10 • 10 = 100.

–10 is also a square root, since –10 • (–10) = 100.

49 = –7–

49 = 7

100 = 10

100 = –10–

A. 49

B. 100The square roots of 49 are ±7.

The square roots of 100 are ±10.

Evaluating Algebraic Expressions

4-6 Squares and Square Roots

A. 25

Check It Out! Example 1

5 is a square root, since 5 • 5 = 25.

–5 is also a square root, since –5 • (–5) = 25.

12 is a square root, since 12 • 12 = 144.

–12 is also a square root, since –12 • (–12) = 144.

25 = –5–

25 = 5

144 = 12

144 = –12–

Find the two square roots of each number.

B. 144

The square roots of 144 are ±12.

The square roots of 25 are ±5.

Evaluating Algebraic Expressions

4-6 Squares and Square Roots

132 = 169

The window is 13 inches wide.

Find the square root of 169 to find the width of the window. Use the positive square root; a negative length has no meaning.

Additional Example 2: Application

A square window has an area of 169 square inches. How wide is the window?

So 169 = 13.

Evaluating Algebraic Expressions

4-6 Squares and Square Roots

Find the square root of 16 to find the width of the table. Use the positive square root; a negative length has no meaning.

Check It Out! Example 2

A square shaped kitchen table has an area of 16 square feet. Will it fit through a van door that has a 5 foot wide opening?

So the table is 4 feet wide, which is less than 5 feet, so it will fit through the van door.

16 = 4

Evaluating Algebraic Expressions

4-6 Squares and Square RootsAdditional Example 3: Finding the Square Root of a

MonomialSimplify the expression.

A.

Write the monomial as a square.

Use the absolute-value symbol.= 12|c|

144c2

144c2 = (12c)2

B. z6

z6 = (z3)2

= |z3|

Write the monomial as a square: z6 = (z3)2

Use the absolute-value symbol.

Evaluating Algebraic Expressions

4-6 Squares and Square RootsAdditional Example 3: Finding the Square Root of a

Monomial

Simplify the expression.

C.

Write the monomial as a square.

10n2 is nonnegative for all values of n. The absolute-value symbol is not needed.

= 10n2

100n4

100n4 = (10n2)2

Evaluating Algebraic Expressions

4-6 Squares and Square RootsCheck It Out! Example 3

Simplify the expression.

A.

Write the monomial as a square.

Use the absolute-value symbol.= 11|r|

121r2

121r2 = (11r)2

B. p8

p8 = (p4)2

= |p4|

Write the monomial as a square: p8 = (p4)2

Use the absolute-value symbol.

Evaluating Algebraic Expressions

4-6 Squares and Square RootsCheck It Out! Example 3

Simplify the expression.

C.

Write the monomial as a square.

9m2 is nonnegative for all values of m. The absolute-value symbol is not needed.

= 9m2

81m4

81m4 = (9m2)2

Evaluating Algebraic Expressions

4-6 Squares and Square RootsLesson Quiz

12 50

7|p3| z4

5. Ms. Estefan wants to put a fence around 3 sides of a square garden that has an area of 225 ft2. How much fencing does she need?45 ft

Find the two square roots of each number.

1. 144 2. 2500

Simplify each expression.

3. 49p6 4. z8

Evaluating Algebraic Expressions

4-6 Squares and Square Roots

Rational NumbersA rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers.

Evaluating Algebraic Expressions

4-6 Squares and Square Roots

Irrational Numbers

• Any indicated square root whose radicand is not a perfect square is an irrational number.

• The numbers √6, √15, and √

• Most numbers that are not perfect squares have square roots that are irrational numbers

Evaluating Algebraic Expressions

4-6 Squares and Square Roots