Numerical Roots & Radicals

Numerical Roots & Radicals
8th Grade Math Numerical Roots & Radicals

Numerical Roots and Radicals
Squares, Square Roots & Perfect Squares Click on topic to go to that section. Squares of Numbers Greater than 20 Simplifying Perfect Square Radical Expressions Approximating Square Roots Rational & Irrational Numbers Radical Expressions Containing Variables Simplifying Non-Perfect Square Radicands Simplifying Roots of Variables Properties of Exponents Solving Equations with Perfect Square & Cube Roots Common Core Standards: 8.NS.1-2; 8.EE.1-2

Squares, Square Roots and Perfect Squares
Return to Table of Contents

Area of a Square The area of a figure is the number of square units needed to cover the figure. The area of the square below is 16 square units because 16 square units are needed to COVER the figure...

Click to see if the answer found with the Area formula is correct!
Area of a Square The area (A) of a square can be found by squaring its side length, as shown below: A = s2 Click to see if the answer found with the Area formula is correct! A = 42 = 4 4 = 16 sq.units The area (A) of a square is labeled as square units, or units2, because you cover the figure with squares... 4 units

What is the area of a square with sides of 5 inches?
1 What is the area of a square with sides of 5 inches? A 16 in2 B 20 in2 C 25 in2 D 30 in2 Answer: C

What is the area of a square with sides of 6 inches?
2 What is the area of a square with sides of 6 inches? A 16 in2 B 20 in2 C 24 in2 Answer: D D 36 in2

If a square has an area of 9 ft2, what is the length of a side?
3 If a square has an area of 9 ft2, what is the length of a side? A 2 ft B 2.25 ft C 3 ft Answer: C D 4.5 ft

What is the area of a square with a side length of 16 in?
4 What is the area of a square with a side length of 16 in? Answer: 256 sq in

What is the side length of a square with an area of 196 square feet?
5 What is the side length of a square with an area of 196 square feet? Answer: 14 ft

When you square a number you multiply it by itself.
52 = = 25 so the square of 5 is 25. You can indicate squaring a number with an exponent of 2, by asking for the square of a number, or by asking for a number squared. What is the square of seven? What is nine squared? 49 81

Make a list of the numbers 1-15 and then square each of them.
Your paper should be set up as follows: Number Square 3 (and so on)

Number Square The numbers in the right column are squares of the numbers in the left column. If you want to "undo" squaring a number, you must take the square root of the number. So, the numbers in the left column are the square roots of the numbers in the right column.

Square Root Square 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 The square root of a number is found by undoing the squaring. The symbol for square root is called a radical sign and it looks like this: Using our list, to find the square root of a number, you find the number in the right hand column and look to the left. So, the = 9 What is 169?

Square Perfect Root Square 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 When the square root of a number is a whole number, the number is called a perfect square. Since all of the numbers in the right hand column have whole numbers for their square roots, this is a list of the first 15 perfect squares.

Find the following. You may refer to your chart if you need to.

6 What is ? 1 Answer: 1

8 What is the square of 15 ? Answer: 225

10 What is 132? Answer: 169

12 What is the square of 18? Answer: 324

13 What is 11 squared? Answer: 121

14 What is 20 squared? Answer: 400

Squares of Numbers Greater than 20

Think about this... What about larger numbers? How do you find ?

What pattern do you notice?
It helps to know the squares of larger numbers such as the multiples of tens. 102 = 100 202 = 400 302 = 900 402 = 1600 502 = 2500 602 = 3600 702 = 4900 802 = 6400 902 = 8100 1002 = 10000 What pattern do you notice? Teacher Instructions: Number in tens place squared, then two zeros are added

For larger numbers, determine between which two multiples of ten the number lies.
102 = = 1 202 = = 4 302 = = 9 402 = = 16 502 = = 25 602 = = 36 702 = = 49 802 = = 64 902 = = 81 1002 = = 100 Next, look at the ones digit to determine the ones digit of your square root.

Ends in nine so square root ends in 3 or 7 Try 53 then 57 532 = 2809
Examples: Lies between 2500 & 3600 (50 and 60) Ends in nine so square root ends in 3 or 7 Try 53 then 57 532 = 2809 Lies between 6400 and 8100 (80 and 90) Ends in 4 so square root ends in 2 or 8 Try 82 then 88 822 = NO! 882 = 7744 2809 Teacher Instructions: List of Squares 102 = = 1 202 = = 4 302 = = 9 402 = = 16 502 = = 25 602 = = 36 702 = = 49 802 = = 64 902 = = 81 1002 = = 100 7744

15 Find. Answer: 28 Teacher Instructions: List of Squares
102 = = 1 202 = = 4 302 = = 9 402 = = 16 502 = = 25 602 = = 36 702 = = 49 802 = = 64 902 = = 81 1002 = = 100

16 Find. 42 Answer: 42 Teacher Instructions: List of Squares
102 = = 1 202 = = 4 302 = = 9 402 = = 16 502 = = 25 602 = = 36 702 = = 49 802 = = 64 902 = = 81 1002 = = 100 42

102 = = 1 202 = = 4 302 = = 9 402 = = 16 502 = = 25 602 = = 36 702 = = 49 802 = = 64 902 = = 81 1002 = = 100

Simplifying Perfect Square
Radical Expressions Return to Table of Contents

Can you recall the perfect squares from 1 to 400?
12 = = = 22 = 92 = = 32 = = = 42 = = = 52 = = = 62 = = = 72 = =

Square Root Of A Number Recall: If b2 = a, then b is a square root of a. Example: If 42 = 16, then 4 is a square root of 16 What is a square root of 25? 64? 100? 5 8 10

Square Root Of A Number Square roots are written with a radical symbol
Positive square root: = 4 Negative square root: = - 4 Positive & negative square roots: = 4 Negative numbers have no real square roots no real roots because there is no real number that, when squared, would equal -16.

Is there a difference between Which expression has no real roots?
& ? Which expression has no real roots? Evaluate the expressions:

Evaluate the expression
is not real

24 Answer: 1

25 ? Answer: 9

26 = ? Answer: 20

27 Answer: 5

28 Answer: 11

29 = ? A 3 B -3 C No real roots Answer: C

The expression equal to is equivalent to a positive integer when b is
30 The expression equal to is equivalent to a positive integer when b is A -10 B 64 C 16 Answer: A D 4 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

Square Roots of Fractions
b = b 0 4 16 49 = = 7

Try These

31 A C Answer: A B D no real solution

32 A C Answer: A B D no real solution

33 A C Answer: C B D no real solution

34 A C Answer: D B D no real solution

35 A C Answer: C B D no real solution

Square Roots of Decimals
Recall:

To find the square root of a decimal, convert the decimal to a fraction first. Follow your steps for square roots of fractions. = .2 = .05 = .3

36 Evaluate A C B D no real solution Answer: C

37 Evaluate A B .06 .6 C 6 D No Real Solution Answer: B

38 Evaluate A .11 B 11 C 1.1 D No Real Solution Answer: A

39 Evaluate A .8 B .08 C D No Real Solution Answer: B

40 Evaluate A B C D No Real Solution Answer: D

Approximating Square Roots

All of the examples so far have been from perfect squares.
What does it mean to be a perfect square? The square of an integer is a perfect square. A perfect square has a whole number square root.

You know how to find the square root of a perfect square.
What happens if the number is not a perfect square? Does it have a square root? What would the square root look like?

Square Perfect Root Square 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 Think about the square root of 50. Where would it be on this chart? What can you say about the square root of 50? 50 is between the perfect squares 49 and 64 but closer to 49. So the square root of 50 is between 7 and 8 but closer to 7.

Square Perfect Root Square 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 When estimating square roots of numbers, you need to determine: Between which two perfect squares it lies (and therefore which 2 square roots). Which perfect square it is closer to (and therefore which square root). Example: 110 Lies between 100 & 121, closer to 100. So is between 10 & 11, closer to 10.

Estimate the following: 30
Square Perfect Root Square Estimate the following: 30 200 215 Teacher Instructions: √30 - Lies between 25 & 36, closer to 25 but since almost right in the middle,  5.5 √ Lies between 196 & 225, closer to 196 so  14 √ Lies between 196 & 225, closer to 225 so  15

Approximating a Square Root
Approximate to the nearest integer < < Identify perfect squares closest to 38 Take square root 6 < < 7 Answer: Because 38 is closer to 36 than to 49, is closer to 6 than to 7. So, to the nearest integer, = 6

Approximate to the nearest integer
Identify perfect squares closest to 70 Take square root Identify nearest integer < Approximate to the nearest integer

Another way to think about it is to use a number line.
Since 8 is closer to 9 than to 4, √8 is closer to 3 than to 2, so √8 ≈ 2.8 √8

Example: Approximate

The square root of 40 falls between which two perfect squares?
41 The square root of 40 falls between which two perfect squares? A 9 and 16 B 25 and 36 C 36 and 49 D Answer: C 49 and 64

Which whole number is 40 closest to?
42 Which whole number is closest to? Identify perfect squares closest to 40 Take square root Identify nearest integer < Answer: 6

The square root of 110 falls between which two perfect squares?
43 The square root of 110 falls between which two perfect squares? A 36 and 49 B 49 and 64 C 64 and 84 D Answer: D 100 and 121

Estimate to the nearest whole number. 110
44 Estimate to the nearest whole number. 110 Answer: 10

What is the square root of 400?
47 What is the square root of 400? Answer: 20

Approximate to the nearest integer.
48 Approximate to the nearest integer. Answer: 5

Approximate to the nearest integer. 96
49 Approximate to the nearest integer. 96 Answer: 10

The expression is a number between
53 The expression is a number between A 3 and 9 B 8 and 9 C 9 and 10 D 46 and 47 Answer: C From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

Rational & Irrational Numbers

Rational & Irrational Numbers
is rational because the radicand (number under the radical) is a perfect square If a radicand is not a perfect square, the root is said to be irrational. Ex:

Sort the following numbers.
24 25 32 36 40 52 64 100 200 225 1225 300 625 1681 3600 Rational Irrational

Rational or Irrational?
54 Rational or Irrational? A Rational B Irrational Answer: A

55 Rational or Irrational? A Rational B Irrational Answer: B

56 Rational or Irrational? A B Irrational Rational Answer: A

57 Rational or Irrational? A B Irrational Rational Answer: A

58 Rational or Irrational? A B Irrational Rational Answer: B

Which is a rational number?
59 Which is a rational number? A B p C D Answer: C From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

60 Given the statement: “If x is a rational number, then is irrational.” Which value of x makes the statement false? A B 2 Answer: D C 3 D 4 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

Radical Expressions Containing Variables

Square Roots of Variables
To take the square root of a variable rewrite its exponent as the square of a power. (x12)2 = x12 = (a8)2 = = a8

Square Roots of Variables
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. By Definition...

Examples

Try These. = |x|5 = |x|13

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

61 Simplify A B C Answer: B D

62 Simplify A B C Answer: D D

63 Simplify A B C Answer: A D

64 Simplify A B C Answer: C D

65 A C B D no real solution Answer: C

Simplifying Non-Perfect Square Radicands

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.

Try These.

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

66 Simplify A B C Answer: A D already in simplified form

67 Simplify A B C Answer: B D already in simplified form

68 Simplify A B C Answer: A D already in simplified form

69 Simplify A B C Answer: D D already in simplified form

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

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)

Express in simplest radical form.

73 Simplify A B C D Answer: A

When. is written in simplest radical form, the result is
When is written in simplest radical form, the result is What is the value of k? 78 A 20 B 10 C 7 Answer: B D 4 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011. From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

When is expressed in simplest form, what is the value of a?
79 When is expressed in simplest form, what is the value of a? A 6 B 2 C 3 Answer: A D 8 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011. From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

Simplifying Roots of Variables

Remember, when working with square roots, an absolute value sign is needed if: the power of the given variable is even and the answer contains a variable raised to an odd power outside the radical Examples of when absolute values are needed:

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 odd power to start.

Example

Simplify Only the y has an odd power on the outside 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.

80 Simplify A B C D Answer: A
Teacher Instructions: Remember the square root of x2 = |x| so the square root of x6 = |x|3 D

81 Simplify A B C Answer: D D

Properties of Exponents

Rules of Exponents Materials Exponential Table Questions.pdf
Exponential Table.pdf Exponential Test Review.pdf There are handouts that can be used along with this section. They are located under the heading labs on the Exponential page of PMI Algebra. Documents are linked. Click the name above of the document.

The Exponential Table x 1 2 3 4 5 6 7 8 9 10 X X X X X X X X X X

Question 1 x 1 2 3 4 5 6 7 8 9 10 16 729 X X X X X X X X X X Why is 24 equivalent to 42? Write the values out in expanded form and see if you can explain why.

Question 2 x 1 2 3 4 5 6 7 8 9 10 16 64 1024 X X X X X X X X X X 16 x 64 = 1024 42 x 43 = 45 Write the equivalent expressions in expanded form. Attempt to create a rule for multiplying exponents with the same base.

Question 3 x 1 2 3 4 5 6 7 8 9 10 27 216 X X X X X X X X X X 8 x 27 = 216 23 x 33 = 63 Write the equivalent expressions in expanded form. Attempt to create a rule for multiplying exponents with the same power.

Question 4 x 1 2 3 4 5 6 7 8 9 10 25 625 15625 X X X X X X X X X X 15625 ÷ 625 = 25 56 ÷ 54 = 52 Write the equivalent expressions in expanded form. Attempt to create a rule for dividing exponents with the same base.

A. 1. Explain why each of the following statements is true.
A. 23 x 22 = 25 (2 x 2 x 2) x (2 x 2) = (2 x 2 x 2 x 2 x 2) B. 34 x 33 = 37 C. 63 x 65 = 68 am x an = am + n

B. 1. Explain why each of the following statements is true.
A. 23 x 33 = 63 (2 x 2 x 2) x (3 x 3 x 3) = (2 x 3)(2 x 3)(2 x 3) B. 53 x 63 = 303 C x 44 = 404 am x bm = (ab)m

C. 1. Explain why each of the following statements is true.
B. 92 = (32)2 = 34 C = (53)2 = 56 (am)n = amn

B. 45 = D. 1. Explain why each of the following statements is true.
46 B. 45 C. 510 510 = 33 41 50 am an = am-n

Operating with Exponents
Examples am x an = am+n 32 x 34 = 36 am x bm = (ab)m 52 x 32 = 152 (43)2 = 46 (am)n = amn am = am-n an 35 = 32 33

84 Simplify: 43 x 45 A 415 B 48 C 42 Answer: B D 47

85 Simplify: 57 ÷ 53 A 52 B 54 C 521 Answer: B D 510

86 Simplify: 47 x 57 A B C Answer: C D

87 Simplify: A B C Answer: B D

88 Simplify: A B C Answer: B D

The expression (x2z3)(xy2z) is equivalent to
89 The expression (x2z3)(xy2z) is equivalent to A x2y2z3 B C x3y3z4 D x4y2z5 Answer: B From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

90 Simplify: A B C Answer: C D

91 Simplify: A B C Answer: A D simplified

The expression is equivalent to
92 The expression is equivalent to A 2w5 B 2w8 C 20w5 Answer: C D 20w8 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

If x = - 4 and y = 3, what is the value of x - 3y2?
93 If x = - 4 and y = 3, what is the value of x - 3y2? A -13 B -23 C -31 Answer: C D -85 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

When -9 x5 is divided by -3x3, x ≠ 0, the quotient is
94 When -9 x5 is divided by -3x3, x ≠ 0, the quotient is A –3x2 B 3x2 C –27x15 Answer: B D 27x8 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

By definition: x-1 = , x 0

Which expression is equivalent to x-4?
95 Which expression is equivalent to x-4? A B x4 C -4x Answer: A D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

96 What is the value of 2-3? A B C -6 D -8
Answer: B D -8 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

Which expression is equivalent to x-1• y2?
97 Which expression is equivalent to x-1• y2? xy2 A C B D xy-2 Answer: B From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

Solving Equations with Perfect Square and Cube Roots

The product of two equal factors is the "square" of the number.
The product of three equal factors is the "cube" of the number.

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. x3 = # you must find the cube root of both sides in order to find the value of x.

Example: Solve. Divide each side by the coefficient. Then take the square root of each side.

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

Try These: Solve. ± 10 ±8 ±9 ±7

Try These: Solve. 2 4 1 5

98 Solve. Answer: 12

Numerical Roots & Radicals

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