Find square roots. Find cube roots. 7.1 Objective The student will be able to:

Find square roots.

Find cube roots.

7.1 ObjectiveThe student will be able to:

If x2 = y then x is a square root of y.In the expression

is the radical sign and64 is the radicand.

1. Find the square root:

8


-0.2

11, -11


21


-5/9


The index of a cube root is always 3.

The cube root of 64 is written as .3 64

Cube Roots

What does cube root mean?

The cube root of a number is…

…the value when multiplied by itself three times gives the original number.

Cube Root Vocabulary

n xindex

radicand

radical sign

If a number is a perfect cube, then you can find its exact cube root.A perfect cube is a number that can be written as the cube (raised to third power) of another number.

Perfect Cubes

What are Perfect Cubes?

•13 = 1 x 1 x 1 = 1•23 = 2 x 2 x 2 = 8•33 = 3 x 3 x 3 = 27•43 = 4 x 4 x 4 = 64•53 = 5 x 5 x 5 = 125•and so on and on and on…..

Examples:

34 4 4 4 64

4643

because

644444 3

because

3 64 4

Examples:

3327

3

5

4

125

64

36216

3273

5

4

125

643

62163

Examples:

33 28 aa

3515 464 yy

3412 327 mm

aa 283 3

53 15 464 yy

43 12 327 mm

Not all numbers or expressions have an exact cube root as in the previous examples.

If a number is NOT a perfect cube, then you might be able to SIMPLIFY it.

Simplify Cube Roots

2 Extract the cube root of the factor that is a perfect cube.

1 Write the radicand as a product of two factors, where one of the factors is a perfect cube.

To simplify a cube root ...

3 The factors that are not perfect cubes will remain as the radicand.

perfect cube

Examples:

3 227 3 23

3 36 3 2125 4 a b ab 3 22 45 abba

3 1043 64 10

3 36 3 23 125 4 a a b b

3 541)

3 6402)

3 7 5500 a b3)

Not all cube roots

can be simplified!

• 30 is not a perfect cube.

• 30 does not have a perfect cube factor.

Example: 3 30

cannot be simplified! 3 30


use the Pythagorean Theorem

What is a right triangle?

It is a triangle which has an angle that is 90 degrees.

The two sides that make up the right angle are called legs.

The side opposite the right angle is the hypotenuse.

leg

leg

hypotenuse

right angle

The Pythagorean Theorem

In a right triangle, if a and b are the measures of the legs and c is the

hypotenuse, then

a2 + b2 = c2.

Note: The hypotenuse, c, is always the longest side.

Find the length of the hypotenuse if1. a = 12 and b = 16.

122 + 162 = c2

144 + 256 = c2

400 = c2

Take the square root of both sides.

20 = c

2400 c

52 + 72 = c2

25 + 49 = c2

74 = c2

Take the square root of both sides.

8.60 = c

Find the length of the hypotenuse if2. a = 5 and b = 7.

274 c

3. Find the length of the hypotenuse given a = 6 and b = 12

1. 180

2. 324

3. 13.42

4. 18

Find the length of the leg, to the nearest hundredth, if

4. a = 4 and c = 10.42 + b2 = 102

16 + b2 = 100Solve for b.

16 - 16 + b2 = 100 - 16b2 = 84

b = 9.17

2 84b

Find the length of the leg, to the nearest hundredth, if5. c = 10 and b = 7.

a2 + 72 = 102

a2 + 49 = 100Solve for a.a2 = 100 - 49

a2 = 51

a = 7.14

2 51a

6. Find the length of the missing side given a = 4 and c = 5

1. 1

2. 3

3. 6.4

4. 9

7. The measures of three sides of a triangle are given below. Determine whether each

triangle is a right triangle. , 3, and 8

Which side is the biggest?

The square root of 73 (= 8.5)! This must be the hypotenuse (c).

Plug your information into the Pythagorean Theorem. It doesn’t matter which number is

a or b.

9 + 64 = 7373 = 73

Since this is true, the triangle is a right triangle!! If it was not true, it

would not be a right triangle.

Sides: , 3, and 832 + 82 = ( ) 2

8. Determine whether the triangle is a right triangle given the sides 6, 9, and 45

1. Yes

2. No

3. Purple

7.3 ObjectivesThe student will be able to:

1. simplify square roots, and

2.simplify radical expressions.

1 • 1 = 12 • 2 = 43 • 3 = 9

4 • 4 = 165 • 5 = 256 • 6 = 36

49, 64, 81, 100, 121, 144, ...

What numbers are perfect squares?

1. Simplify

Find a perfect square that goes into 147.

147 7 3

2. Simplify


11 5

Simplify

1. .

2. .

3. .

4. .

2 18

72

3 8

6 236 2

Look at these examples and try to find the pattern…

How do you simplify variables in the radical?

1x x2x x3x x x4 2x x5 2x x x6 3x x

What is the answer to ?

7 3x x x

As a general rule, divide the exponent by two. The remainder stays in the

radical.


4. Simplify

7x

5. Simplify 258x

122 2x x

Simplify 369x

1. 3x6

2. 3x18

3. 9x6

4. 9x18

Multiply the radicals.

7. Simplify

60

4 154 152 15

8. Simplify Multiply the coefficients and radicals.

6 294

6 49 66 649

42 6

6 67

9.Simplify

1. .

2. .

3. .

4. .

24 3x44 3x

2 48x448x

36 8x x

How do you know when a radical problem is done?

1. No radicals can be simplified.Example:

2. There are no fractions in the radical.Example:

3. There are no radicals in the denominator.Example:

8

1

4

1

5

10. Simplify.

Divide the radicals.

108

3

366

Uh oh…There is a

radical in the denominator!

Whew! It simplified!

11. Simplify

4 1

4

4

2

2

Uh oh…Another

radical in the denominator!

Whew! It simplified again! I hope they all are like this!

12. Simplify

5

7

5

7

75

7 7

35

49 35

7

Since the fraction doesn’t reduce, split the radical up.

Uh oh…There is a fraction in the radical!

How do I get rid of the radical in

the denominator?

Multiply by the “fancy one” to make the denominator a

perfect square!


simplify radical expressions involving addition and subtraction.

1. Simplify.

Just like when adding variables, you can only combine LIKE radicals.

5 5

Which are like radicals?

2. Simplify.

4 7 3 3

Simplify

1. .

2. .

3. .

4. .

5 2 6 2 4 2

5 2 6 2 4 2 15 2

3 2

7 2

Perimeter = Add all of the sides

3. Find the perimeter of a rectangle whose length is and whose

width is

8 6 6 3 8

Simplify each radical.

4. Simplify.

9 164 3 2 3 42 5

3 44 3 2 3 22 5

12 3 8 3 4 5

4 3 4 5

Combine like radicals.

5. Simplify

25 368 2 5 2 2 249 5 68 2 5 2 72 2

40 2 30 2 14 2 56 2

Simplify

1. .

2. .

3. .

4. .

5 3 4 2 3 3

5 3 4 2 3 3 6 22 3 4 28 3 4 2

Simplify

1. .

2. .

3. .

4. .

3 12 4 27

7 39

48 3

48 6

18 3

Find square roots. Find cube roots. 7.1 Objective The student will be able to:

Documents