8th Grade Equations with Roots and Radicals 2015-12-17 - - PDF document

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8th Grade

Equations with Roots and Radicals

2015-12-17 www.njctl.org

Slide 3 / 87 Table of Contents

Radical Expressions Containing Variables Simplifying Non-Perfect Square Radicands Simplifying Roots of Variables

Click on topic to go to that section.

Solving Equations with Perfect Square & Cube Roots Glossary & Standards

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Radical Expressions Containing Variables

Return to Table of Contents

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To take the square root of a variable rewrite its exponent as the square of a power.

Square Roots of Variables

= = (x12)2 = x

12

(a8)2 = a

8

Can you find a shortcut to solve this type of problem? How would your shortcut make the problem easier?

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If the square root of a variable raised to an even power has a variable raised to an odd power for an answer, the answer must have absolute value signs. This ensures that the answer will be positive.

Square Roots of Variables

By Definition...

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Examples

Square Root Practice Slide 8 / 87

Try These.

= |x|

5

= |x|

13

Square Root Practice Slide 9 / 87

no no How many of these expressions will need an absolute value sign when simplified? yes yes yes yes

Square Root Practice Slide 10 / 87 Slide 11 / 87 Slide 12 / 87

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5 A B C D no real solution

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Simplifying Non-Perfect Square Radicands

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A number is a perfect square if you can take that quantity of 1x1 unit squares and form them into a square. 1 1

Unit Square

4 is a perfect square, because you can take 4 unit squares and form them into a 2x2 square. (Notice that the square root of 4 is the length of one

• f its sides, since that side times itself equals 4.)

2 2

4 = 2

Simplifying Perfect Squares

(Review)

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What About Numbers that are not Perfect Squares? How can we simplify ? 8 8 is not a perfect square, and no matter how we arrange the square units, we will not be able to form them into a square. So, we know that we will not have a whole number, which we can multiply by itself, to equal 8.

Non-Perfect Squares Slide 18 / 87

What happens when the radicand is not a perfect square? Rewrite the radicand as a product of its largest perfect square factor. Simplify the square root of the perfect square. When simplified form still contains a radical, it is said to be irrational. =

22 2

8

8

click click

Non-Perfect Squares

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What happens when the radicand is not a perfect square?

• 1. Rewrite the radicand as a product of its largest perfect

square factor.

• 2. Simplify the square root of the perfect square.

When simplified form still contains a radical, it is said to be irrational.

click click

Non-Perfect Squares

click

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Identifying the largest perfect square factor when simplifying radicals will result in the least amount of work. Ex: Not simplified! Keep going! Finding the largest perfect square factor results in less work: Note that the answers are the same for both solution processes

Simplifying Non-Perfect Squares Slide 21 / 87 Simplifying Non-Perfect Squares

Another method for simplifying non-perfect squares is to use prime factorization and a factor tree. For example, 48 can be broken down as follows: 48 2 24 2 12 2 6 2 3

Slide 22 / 87 Simplifying Non-Perfect Squares

After you factor the number into all of its primes, you can circle each pair of numbers that exist to signify that they come outside of the radical. For each pair circled, one number comes out. If more than one pair of numbers are circled, join the numbers outside of the radical by a multiplication sign. Any numbers left without a match must stay inside of the radical. Multiply them together, if

• needed. Therefore, 48 simplifies to 4 3.

48 2 24 2 12 2 6 2 3 2(2) 3 = 4 3

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Try These.

Non-Perfect Squares Practice Slide 24 / 87

6 Simplify A B C D already in simplified form

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7 Simplify A B C D already in simplified form

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8 Simplify A B C D already in simplified form

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9 Simplify A B C D already in simplified form

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10 Simplify A B C D already in simplified form

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11 Simplify A B C D already in simplified form

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12 Which of the following does not have an irrational simplified form? A B C D

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13 The diagonal of a square can be expressed by the formula d= 2a2, where a is the side length of the square. Select the correct options to show the length of the diagonal of the square shown. Your answer should be a radicand in simplest form. A 3 B 4 C 9 D 1 E 2 F 3

d = ___ ___

9

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14 The distance, d, in miles that a person can see to the horizon is calculated with the following formula. A 3 B 4 C 5 D 5 E 6 F 10

d = ___ ___

3h 2 d = h = the person's height above sea level in feet.

100 ft above sea level

How far to the horizon would you be able to see from this vantage point? Your answer should be a radicand in simplest form.

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2 Note - If a radical begins with a coefficient before the radicand is simplified, any perfect square that is simplified will be multiplied by the existing coefficient. (multiply the outside)

Simplest Radical Form Slide 34 / 87 Simplest Radical Form

Likewise - If a radical begins with a coefficient before the radicand is simplified, any pair of primes that are circled will be multiplied by the existing coefficient. (multiply the outside) 2 18 2 9 3 3 2(3) 2 6 2 7 12 2 6 2 3 7(2) 3 14 3

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15 Simplify A B C D

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16 Simplify A B C D

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17 Simplify A B C D

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18 Simplify A B C D

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19 Simplify A B C D

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Teachers: Use the questions found in the pull tab for the next 2 slides.

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20 When is written in simplest radical form, the result is . What is the value of k? A 20 B 10 C 7

From the New York State Education Department. Office of Assessment Policy, Development and

• Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

D 4

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21 When is expressed in simplest form, what is the value of a? A 6 B 2 C 3

From the New York State Education Department. Office of Assessment Policy, Development and

• Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

D 8

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22 Which is greater or 6?

Derived from

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23 Which is greater or 10?

Derived from

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Simplifying Roots of Variables

Return to Table of Contents

Slide 47 / 87 Using Absolute Value

When we simplify radicals, we are told to assume all variables are positive. But, why? Because, the square root of the square of a negative number is not the original number.

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Take -2 for example. But, 4 is not -2, it is +2. By definition square roots of numbers are positive. You started with a negative number (-2), and ended up with a positive number (+2). So, the square root of a number is the absolute value

• f the square root.

(-2)2 = +4 4 = 2 This accounts for +22 and (-2)2.

Using Absolute Value

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Easy enough. But what about when the radicand is a variable, and we don't know the sign of the unknown value?

x2

Is x positive or negative? We can't know, so we "assume all variables are positive".

Using Absolute Value Slide 50 / 87 Simplifying Roots of Variables

The technical definition of "the square root of x squared" is "the absolute value of x".

x2 = x x2

=

x x

x is positive

x2

=

x x

x is negative

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Simplifying Roots of Variables

Using Absolute Values · The power of the given variable is even. · The answer contains a variable raised to an

• dd power outside the radical.

and When working with square roots, an absolute value sign is needed if:

x6 x3

x6

= x3

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x6

= x3

x x x x x x

Whether x is positive or negative, when it is multiplied by itself an even number of times, it will turn

• ut to be a positive number.

So, x is positive.

= x x x

However, if x is negative, when it is multiplied by itself an odd number of times, it will turn out to be a negative number. So, x could be negative.

So, in order for , we must use an absolute value sign to indicate that x is positive. x6 = x3

x6 = x 3

But, Why? Slide 53 / 87

More Examples Use expanded form to explain why absolute value must be used in these answers.

Roots of Variable Practice Slide 54 / 87 Simplifying Roots of Variables

Divide the exponent by 2. The number of times that 2 goes into the exponent becomes the power on the outside of the radical and the remainder is the power of the radicand. Note: Absolute value signs are not needed because the radicand had an

• dd power to start.

=

x7

x x x x x x x

x3 x

=

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Examples:

50x4y12z

3

25 2(x2)2(y6)2

5 x 2 y 6z 2z z z z

Combining it all:

Roots of Variables Examples Slide 56 / 87

Only the y has an odd power on the

• utside of the radical.

The x had an odd power under the radical so no absolute value signs needed. The m's starting power was odd, so it does not require absolute value signs.

Roots of Variables Practice Slide 57 / 87 Slide 58 / 87 Slide 59 / 87

26 Simplify A B C D

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27 Simplify A B C D

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Solving Equations with Perfect Square and Cube Roots

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(Problem from )

Use the numbers shown to make the equations true. Each number can be used only once. 4 8 10 64 1000 100 a. = = b. 3

Squares and Cubes Practice Slide 64 / 87

Perfect Squares Perfect Cubes Complete the Venn-Diagram to classify the numbers as perfect squares and perfect cubes.

(Problem from )

1 64 96 125 200 256 333 361

Squares and Cubes Practice Slide 65 / 87

When we solve equations, the solution sometimes requires finding a square or cube root of both sides of the equation. When your equation simplifies to: x2 = # you must find the square root of both sides in order to find the value of x. When your equation simplifies to: x3 = # you must find the cube root of both sides in order to find the value of x.

Solving Equations Slide 66 / 87

Example: Solve.

Divide each side by the

• coefficient. Then take the

square root of each side. =

Solving Equations Example

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Example: Solve.

Multiply each side by nine, then take the cube root of each side.

Solving Equations Example Slide 68 / 87 Notice!

The answer is only a positive 3, not 3. +

• Why is the answer only positive and not

both positive and negative?

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The cube root of 27 is 3, and not -3, because when 3 is cubed you get 27. 3 x 3 x 3 = 27 If you were to cube -3, you would get -27...

• 3 x -3 x -3 = -27

Therefore, the cube root of -27 is -3. So we can take a cube root of a positive number AND take the cube root of a negative number!

Cube Roots Slide 70 / 87 Cube Roots Examples Slide 71 / 87

Try These: Solve. ± 10 ± 8 ± 9 ± 7

Squares and Cubes Practice Slide 72 / 87

Try These: Solve. 2 1 4 5

Squares and Cubes Practice

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28 Solve.

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29 Solve.

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30 Solve.

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31 Solve.

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32 Solve 15 + x2 = 40

Derived from

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33 Solve 2 + x3 = 10

Derived from

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34 A cube has a volume of 343 cm3. a) Write an equation that could be used to determine the length, L, of

• ne side.

b) Solve the equation.

Derived from

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35 Estimate the area of the rectangle to the nearest tenth.

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36 If the area of a square is square inches, what is the length, in inches, of one side of the square? A B C D

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37 Which equation has both 4 and -4 as possible values of y?

From PARCC EOY sample test non-calculator #9

A B C D

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Glossary & Standards

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Back to Instruction

Cube

To multiply a number by itself and then again by itself. The product of three equal factors. 43 =

4 x 4 x 4 =

64

What is 4 cubed? (4)(4)(4) =

63 =

6 x 6 x 6 =

216

What is the cube of 6? (6)(6)(6) =

103 =

10 x 10 x 10 =

1000

What is 10 cubed? (10)(10)(10) =

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Back to Instruction

Cube Root

A value that, when used in a multiplication three times, gives that number.

3

Symbol:

"cube root"

3 3

64 = 4

(4)(4)(4) = 64 4x4x4 = 64

216 = 6

(6)(6)(6) = 216 6x6x6 = 216

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Power

3

2

Base

Power

3

2= x

3 3

3 = x x 3 3 3

3

3

2 x

2 3 3

3 x

3 3

"3 to the second power"

A power is another name for an exponent. It is a small, raised number that shows how many times to multiply the base by itself.

Back to Instruction

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MP8 Look for & express regularity in repeated reasoning.

Click on each standard to bring you to an example of how to meet this standard within the unit.

Standards for Mathematical Practice

MP1 Making sense of problems & persevere in solving them. MP2 Reason abstractly & quantitatively. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP5 Use appropriate tools strategically. MP6 Attend to precision. MP7 Look for & make use of structure.