algebra1 radical expressions

CONFIDENTIAL 1

Algebra1Algebra1

RadicalRadicalExpressionsExpressions

CONFIDENTIAL 2

Warm UpWarm Up

1) {(-3, 16), (-2, 8), (0, 2), (1, 1), (3, 0.25)}

2) {(-5, 15), (-2, -6), (0, -10), (3, -1), (4, 6)}

Graph each data set. Which kind of model best describes the data?

CONFIDENTIAL 3

Radical ExpressionsRadical Expressions

An expression that contains a radical sign (√) is a radical expression . There are many different types of radical

expressions, but in this course, you will only study radicalexpressions that contain square roots.

The expression under a radical sign is the radicand . A radicand may contain numbers, variables, or both. It may

contain one term or more than one term.

Examples of radical expressions:

14 l2 + w2 2gd d 5√2 18 4

CONFIDENTIAL 4

Simplest Form of a Square-Root ExpressionSimplest Form of a Square-Root Expression

An expression containing square roots is in simplest form when

• the radicand has no perfect square factors other than 1.

• the radicand has no fractions.

• there are no square roots in any denominator.

CONFIDENTIAL 5

Remember that positive numbers have two square roots, one positive and one negative. However, √1 indicates a non-negative square root. When you simplify, be sure that your answer is not negative. To simplify √x2 , you

should write √x2 = |1| , because you do not know whether x is positive or negative.

Below are some simplified square-root expressions:√x2 = |x|

√x3 = x√x

√x4 = x2

√x5 = x2√x

√x6 = |x3|

CONFIDENTIAL 6

Simplifying Square-Root ExpressionsSimplifying Square-Root Expressions

Simplify each expression.

A) 2 = 1 = 1 72 36 6

B) 32 + 42 = 9 + 16 = 25 = 5

C) x2 + 8x + 16 = (x + 4)2 = |x + 4|

CONFIDENTIAL 7

Now you try!

Simplify each expression.

1a) 256 4

1b) 40 + 9

1c) 52 + 122

1d) (3 - x)2

CONFIDENTIAL 8

Product Property of Square RootsProduct Property of Square Roots

For any nonnegative real numbers a and b, the square root of ab is equal to the square root of

a times the square root of b.

WORDS

NUMBERS

ALGEBRA

CONFIDENTIAL 9

Using the Product Property of Square RootsUsing the Product Property of Square Roots

Simplify. All variables represent nonnegative numbers.

Factor the radicand using perfect squares.

Product Property of Square Roots

Product Property of Square Roots

Simplify.

Product Property of Square Roots

Since y is nonnegative, √y2 = y.

A) 18 = 9(2) = 9 (2)

= 3 (2)

B) x4y3 = x4 (y3) = x4 y2 y

= x2y y

CONFIDENTIAL 10

Simplify. All variables represent nonnegative numbers.

Now you try!

2a) 128

2b) x3 y2

2c) 48a2b

CONFIDENTIAL 11

Quotient Property of Square RootsQuotient Property of Square Roots

For any real numbers a and b (a ≥ 0 and b > 0) , the square root of a is equal to the

b square root of a divided by the square root of b.

WORDS

NUMBERS

ALGEBRA

CONFIDENTIAL 12

Using the Quotient Property of Square RootsUsing the Quotient Property of Square Roots

Simplify. All variables represent nonnegative numbers.

Quotient Property of Square Roots.

Simplify.

A) 5 = 5 9 9 = 5 3

Quotient Property of Square Roots.

Simplify.

B) a5 = a4

81a 81 = a4

81

= a2

9

Simplify.

CONFIDENTIAL 13

Simplify. All variables represent nonnegative numbers.

Now you try!

3a) 12 27

3b) 36 x4

3c) y6

4

CONFIDENTIAL 14

Using the Product and Quotient Using the Product and Quotient Properties TogetherProperties Together

Simplify. All variables represent nonnegative numbers.

Quotient Property

Write 80 as 16 (5) .

Product Property

Simplify.

a) 80 25

= 80 25

= 16(5) 25

= 16 (5) 25

= 4 (5) 5

CONFIDENTIAL 15

Quotient Property

Write 80 as 16 (5) .

Product Property

Simplify.

b) 4x5

9

= 4x5

9

= 4(x5) 9

= 4 (x4) (x) 9

= 4x4 (5) 3

CONFIDENTIAL 16

Simplify. All variables represent nonnegative numbers.

Now you try!

4a) 20 49

4b) z5

25y2

4c) p6

q10

CONFIDENTIAL 17

Sports ApplicationSports Application

A baseball diamond is a square with sides of 90 feet. How far is a throw from third base to first base? Give the

answer as a radical expression in simplest form. Then estimate the length to the nearest tenth of a foot.

The distance from third base to first base is the hypotenuse of a

right triangle.Use the Pythagorean Theorem:

c2 = a2 + b2

Solve for c.

Substitute 90 for a and b.

c = a2 + b2

c = 902 + 902

CONFIDENTIAL 18

Factor 16,200 using perfect squares.

Simplify.

Use the Product Property of Square Roots.

Use a calculator and round to the nearest tenth.

c = 8100 + 8100

c = 16,200

c = 100(81)(2)

c = 100 (81) (2)

c = 10 (9) (2)

c = 90 (2)

c ≈ 127.3

Simplify.

The distance is 90√2 , or about 127.3, feet.

CONFIDENTIAL 19

5) A softball diamond is a square with sides of 60 feet. How long is a throw from third base to first base in softball? Give the answer as a radical expression in simplest form. Then estimate the length to the

nearest tenth of a foot.

Now you try!

CONFIDENTIAL 20

Assessment

1) In the expression, 3x - 6 + 7, what is the radicand ?

2) Your boat is traveling due north from a dock. Your friend’s boat left at the same time from the same dock and is headed due east.

After an hour, your friend calls and tells you that he has just stopped because of engine trouble. How far must you travel to meet your friend? Give your answer as a radical expression in simplest form. Then estimate the distance to the nearest mile.

CONFIDENTIAL 21

Simplify. All variables represent nonnegative numbers.

3) 81

4) 98 2

5) (a + 7)2

6) 180

CONFIDENTIAL 22

Graph each square-root function.

7) 17 25

8) 7 16

9) 108 49

10) 204 25

CONFIDENTIAL 23

Radical ExpressionsRadical Expressions

An expression that contains a radical sign (√) is a radical expression . There are many different types of radical

expressions, but in this course, you will only study radicalexpressions that contain square roots.

The expression under a radical sign is the radicand . A radicand may contain numbers, variables, or both. It may

contain one term or more than one term.

Examples of radical expressions:

14 l2 + w2 2gd d 5√2 18 4

Let’s review

CONFIDENTIAL 24

Simplest Form of a Square-Root ExpressionSimplest Form of a Square-Root Expression

An expression containing square roots is in simplest form when

• the radicand has no perfect square factors other than 1.

• the radicand has no fractions.

• there are no square roots in any denominator.

CONFIDENTIAL 25

Remember that positive numbers have two square roots, one positive and one negative. However, √1 indicates a non-negative square root. When you simplify, be sure that your answer is not negative. To simplify √x2 , you

should write √x2 = |1| , because you do not know whether x is positive or negative.

Below are some simplified square-root expressions:√x2 = |x|

√x3 = x√x

√x4 = x2

√x5 = x2√x

√x6 = |x3|

CONFIDENTIAL 26

Simplifying Square-Root ExpressionsSimplifying Square-Root Expressions

Simplify each expression.

A) 2 = 1 = 1 72 36 6

B) 32 + 42 = 9 + 16 = 25 = 5

C) x2 + 8x + 16 = (x + 4)2 = |x + 4|

CONFIDENTIAL 27

Product Property of Square RootsProduct Property of Square Roots

For any nonnegative real numbers a and b, the square root of ab is equal to the square root of

a times the square root of b.

WORDS

NUMBERS

ALGEBRA

CONFIDENTIAL 28

Using the Product Property of Square RootsUsing the Product Property of Square Roots

Simplify. All variables represent nonnegative numbers.

Factor the radicand using perfect squares.

Product Property of Square Roots

Product Property of Square Roots

Simplify.

Product Property of Square Roots

Since y is nonnegative, √y2 = y.

A) 18 = 9(2) = 9 (2)

= 3 (2)

B) x4y3 = x4 (y3) = x4 y2 y

= x2y y

CONFIDENTIAL 29

Quotient Property of Square RootsQuotient Property of Square Roots

For any real numbers a and b (a ≥ 0 and b > 0) , the square root of a is equal to the

b square root of a divided by the square root of b.

WORDS

NUMBERS

ALGEBRA

CONFIDENTIAL 30

Using the Quotient Property of Square RootsUsing the Quotient Property of Square Roots

Simplify. All variables represent nonnegative numbers.

Quotient Property of Square Roots.

Simplify.

A) 5 = 5 9 9 = 5 3

Quotient Property of Square Roots.

Simplify.

B) a5 = a4

81a 81 = a4

81

= a2

9

Simplify.

CONFIDENTIAL 31

Using the Product and Quotient Using the Product and Quotient Properties TogetherProperties Together

Simplify. All variables represent nonnegative numbers.

Quotient Property

Write 80 as 16 (5) .

Product Property

Simplify.

a) 80 25

= 80 25

= 16(5) 25

= 16 (5) 25

= 4 (5) 5

CONFIDENTIAL 32

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