1. Write each fraction as a decimal.
Warm Up
Simplify each expression.
1. 62
2. 112
121 36 25 36
3. (–9)(–9) 81 4.
Write each fraction as a decimal. 2 5 5 9 5.
0.4 6.
0.5 5 3 8 –1 5 6 7.
5.375 8.
–1.83

2. Roots and Irrational Numbers
Section 1.5

3. California
Standards
2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.

4. Objectives: In this lesson you’ll:
Evaluate expressions containing roots.
Classify numbers within the real number system 4

5. Words to know…
Square root - a number which, when multiplied by itself, produces the given number. (Ex. 7² = 49, 7 is the square
root of 49)
Perfect square- any number that has an integer square root.(ex. 100 is a perfect square ,
Cube root - a number that is raised to the third power to form a product is a cube root. (ex 23=8, =2)

6. Square Roots Perfect Square Roots Squares 0² = 0 1² = 1 2² = 4 3² = 9
4² = 16
5² = 25
6² = 36
7² = 49
8² = 64
9² = 81
10² = 100
Perfect Square Roots

7. Are squares and square roots inverses?
Start
Square it
Root it
Result 3 3 5 5 9 9
A square root is the inverse operation of a square!

8. Do you know your perfect squares?
7 and -7 25
8 and -8
121
196
3 and -3

9. Square Roots Find the square roots of 16. Solution
Positive real numbers have two square roots.
Find the square roots of 16.
Solution
4  4 = 42 = 16
= 4
Positive square
root of 16
(–4)(–4) = (–4)2 = 16 –
= –4
Negative square
root of 16
The square roots of 16 are 4 and - 4.

10. Writing Math
The small number to the left of the root is the index. In a square root, the index is understood to be 2. In other words, is the same as .
Cube roots
A number that is raised to the third power to form a product is a cube root of that product. The symbol indicates a cube root. Since 23 = 8,
= 2. Similarly, the symbol indicates a fourth root: 2 = 16, so = 2.

11. You try Find each root. Think: What number squared equals 81?
Think: What number cubed equals –216?
(–6)(–6)(–6) = 36(–6) = –216
= –6
= –6
(–6)(–6)(–6) = 36(–6) = –216

12. You try Finding Roots of Fractions. Think: What number squared equals
Think: What number cubed equals b.

13. You try Finding Roots of Fractions. Think: What number squared equals
Think: What number cubed equals B.
= –6
(–6)(–6)(–6) = 36(–6) = –216

14. Approximating Square Roots
Square roots of numbers that are not perfect squares, such as 15, are not whole numbers. A calculator can approximate the value of as Without a calculator, you can use square roots of perfect squares to help estimate the square roots of other numbers.
If a whole number is not a perfect square, then its square root is irrational. For example, 2 is not a perfect square and is irrational.
Remember

15. Approximating Square Roots
Approximate to the nearest whole number.
Solution
Is between 7² and 8².

16. Let’s practice… Between 2 and 3 Between 4 and 5 Between 4 and 5
Determine what two consecutive integers each root lies between.
Between 2 and 3
Between 4 and 5
Between 4 and 5
Between 5 and 6

17. Words to know…
Natural numbers - The counting numbers. (example: 1, 2, 3…)
Whole numbers - The natural numbers and zero.(example: 0, 1,2,3…)
Integers -The whole numbers and their opposites.(ex: …-3,-2,-1,0,1,2,3…)
Rational numbers - Numbers that can be expressed as a fraction (a/b).

18. Words to know…
Terminating decimal -Rational numbers in decimal form that have finite (ends) number of digits. (ex 2/5= 0.40 )
Repeating decimal -rational numbers in decimal form that have a block for one or more digits that repeats continuously. (ex. 1.3= )
Irrational numbers - numbers that cannot be expressed as a fraction including square roots of whole numbers that are not perfect squares and nonterminating decimals that do not repeat.

19. The real numbers are made up of all rational and irrational numbers.
Note the symbols for the sets of numbers.
R: real numbers
Q: rational numbers
Z: integers
W: whole numbers
N: natural numbers
Reading Math

20. Classifying Real Numbers
Write all classifications that apply to each real number. A.
–32 32 1
–32 can be written in the form .
–32 = –
–32 can be written as a terminating decimal.
–32 = –32.0
rational number, integer, terminating decimal B.
14 is not a perfect square, so is irrational.
irrational

21. Write all classifications that apply to each real number.
Check It Out!
Write all classifications that apply to each real number.
7 can be written in the form .
4 9
a. 7
can be written as a repeating decimal.
67  9 = 7.444… = 7.4
rational number, repeating decimal
b. –12
–12 can be written in the form .
–12 can be written as a terminating decimal.
rational number, terminating decimal, integer

22. Write all classifications that apply to each real number.
10 is not a perfect square, so
is irrational.
irrational
100 is a perfect square, so is rational.
10 can be written in the form and as a terminating decimal.
natural, rational, terminating decimal, whole, integer

23. A challenge…
Would you know how to solve this….
-11
-11
x = 5

24. A challenge…
Solve the variable. -3 -3
x = 8

25. Lesson Quiz
Find each square root. 3. 5 4. 1. 3 2. 1
5. The area of a square piece of cloth is 68 in2. Estimate to the nearest tenth the side length of the cloth.
 8.2 in.
Write all classifications that apply to each real number.
6. –3.89
rational, repeating decimal 7.
irrational