Understanding Roots in Mathematics

1. Preview Warm Up California Standards Lesson Presentation

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

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. Vocabulary square root terminating decimal principal square root repeating decimal perfect square irrational numbers cube root natural numbers whole numbers integers rational numbers

5. A number that is multiplied by itself to form a product is a square root of that product. The radical symbol is used to represent square roots. For nonnegative numbers, the operations of squaring and finding a square root are inverse operations. In other words, for x ≥ 0, Positive real numbers have two square roots. Positive square root of 16 =4 4  4 = 42= 16 = –4 – (–4)(–4) = (–4)2= 16 Negative square root of 16

6. The principal square root of a number is the positive square root and is represented by . A negative square root is represented by – . The symbol is used to represent both square roots. A perfect square is a number whose positive square root is a whole number. Some examples of perfect squares are shown in the table. 0 1 4 9 16 25 36 49 64 81 100 02 12 22 32 42 52 62 72 82 92 102

7. 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 .

8. 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.

9. Additional Example 1: Finding Roots Find each root. Think: What number squared equals 81? Think: What number squared equals 25?

10. Additional Example 1: Finding Roots Find the root. C. Think: What number cubed equals –216? (–6)(–6)(–6) = 36(–6) = –216 = –6

11. a. b. Check It Out! Example 1 Find each root. Think: What number squared equals 4? Think: What number squared equals 25?

12. Check It Out! Example 1 Find the root. c. Think: What number to the fourth power equals 81?

13. Think: What number squared equals Additional Example 2: Finding Roots of Fractions Find the root. A.

14. Think: What number cubed equals Additional Example 2: Finding Roots of Fractions Find the root. B.

15. Think: What number squared equals Additional Example 2: Finding Roots of Fractions Find the root. C.

16. Think: What number squared equals Check It Out! Example 2 Find the root. a.

17. Think: What number cubed equals Check It Out! Example 2 Find the root. b.

18. Think: What number squared equals Check It Out! Example 2c Find the root. c.

19. Square roots of numbers that are not perfect squares, such as 15, are not whole numbers. A calculator can approximate the value of as 3.872983346... Without a calculator, you can use square roots of perfect squares to help estimate the square roots of other numbers.

20. Since the area of the square is 13 in², then each side of the square is in. 13 is not a perfect square, so find two consecutive perfect squares that is between: 9 and 16. is between and , or 3 and 4. Refine the estimate. Additional Example 3: ArtApplication As part of her art project, Shonda will need to make a paper square covered in glitter. Her tube of glitter covers 13 in². Estimate to the nearest tenth the side length of a square with an area of 13 in².

21. Since 3.6 is too low and 3.65 is too high, is between 3.6 and 3.65. Round to the nearest tenth. The side length of the paper square is Additional Example 3 Continued 3.5 3.52 = 12.25 too low 3.6 3.62 = 12.96 too low 3.65 3.652 = 13.32 too high

22. Writing Math The symbol ≈ means “is approximately equal to.”

23. Since the area of the square is 26 ft², then each side of the square is ft. 26 is not a perfect square, so find two consecutive perfect squares that is between: 25 and 36. is between and , or 5 and 6. Refine the estimate. Check It Out! Example 3 What if…? Nancy decides to buy more wildflower seeds and now has enough to cover 26 ft2. Estimate to the nearest tenth the side length of a square garden with an area of 26 ft2.

24. Since 5.0 is too low and 5.1 is too high, is between 5.0 and 5.1. Rounded to the nearest tenth,  5.1. The side length of the square garden is  5.1 ft. Check It Out! Example 3 Continued 5.0 5.02 = 25 too low 5.1 5.12 = 26.01 too high

25. Real numbers can be classified according to their characteristics. Natural numbers are the counting numbers: 1, 2, 3, … Whole numbers are the natural numbers and zero: 0, 1, 2, 3, … Integers are the whole numbers and their opposites: –3, –2, –1, 0, 1, 2, 3, …

26. Rational numbers are numbers that can be expressed in the form , where a and b are both integers and b ≠ 0. When expressed as a decimal, a rational number is either a terminating decimal or a repeating decimal. • A terminating decimal has a finite number of digits after the decimal point (for example, 1.25, 2.75, and 4.0). • A repeating decimal has a block of one or more digits after the decimal point that repeat continuously (where all digits are not zeros).

27. Irrational numbersare all numbers that are not rational. They cannot be expressed in the form where a and b are both integers and b ≠ 0. They are neither terminating decimals nor repeating decimals. For example: 0.10100100010000100000… After the decimal point, this number contains 1 followed by one 0, and then 1 followed by two 0’s, and then 1 followed by three 0’s, and so on. This decimal neither terminates nor repeats, so it is an irrational number.

28. 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.

29. The real numbers are made up of all rational and irrational numbers.

30. Reading Math Note the symbols for the sets of numbers. R: real numbers Q: rational numbers Z: integers W: whole numbers N: natural numbers

31. –32 can be written in the form . 14 is not a perfect square, so is irrational. Additional Example 4: Classifying Real Numbers Write all classifications that apply to each real number. A. –32 32 1 –32 = – –32 can be written as a terminating decimal. –32 = –32.0 rational number, integer, terminating decimal B. irrational

32. 7 can be written in the form . 67  9 = 7.444… = 7.4 49 can be written as a repeating decimal. –12 can be written in the form . Check It Out! Example 4 Write all classifications that apply to each real number. a. 7 rational number, repeating decimal b. –12 –12 can be written as a terminating decimal. rational number, terminating decimal, integer

33. 10 is not a perfect square, so is irrational. 100 is a perfect square, so is rational. 10 can be written in the form and as a terminating decimal. Check It Out! Example 4 Write all classifications that apply to each real number. irrational natural, rational, terminating decimal, whole, integer

34. 6. –3.89 Lesson Quiz Find each square root. 4. 3. 5 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. 7. rational, repeating decimal irrational