holt mcdougal algebra 1 1-5 roots and real numbers 1-5 roots and real numbers holt algebra 1 lesson...
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Holt McDougal Algebra 1
1-5 Roots and Real Numbers1-5 Roots and Real Numbers
Holt Algebra 1
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Warm UpWarm Up
Holt McDougal Algebra 1
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
Warm UpSimplify each expression.
1. 62 36 2. 112 121
3. (–9)(–9) 81 4. 2536
Write each fraction as a decimal.
5. 25
596.
7. 5 38
8. –1 56
0.4
5.375
0.5
–1.83
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
Evaluate expressions containing square roots.
Classify numbers within the real number system.
Objectives
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
square root terminating decimalprincipal square root repeating decimalperfect square irrational numberscube root natural numberswhole numbersintegersrational numbers
Vocabulary
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
Positive real numbers have two square roots. 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 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,
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
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
02
1
12
1004
22
9
32
16
42
25
52
36
62
49
72
64
82
81
92 102
4 4 = 42 = 16 = 4 Positive squareroot of 16
(–4)(–4) = (–4)2 = 16 = –4 Negative squareroot of 16
–
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
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
02
1
12
1004
22
9
32
16
42
25
52
36
62
49
72
64
82
81
92 102
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.
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
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 .
Writing Math
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
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: 24 = 16, so = 2.
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
Example 1: Finding Roots
Find each root.
Think: What number squared equals 81?
Think: What number squared equals 25?
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
Find the root.
Think: What number cubed equals –216?
Additional Example 1: Finding Roots
= –6 (–6)(–6)(–6) = 36(–6) = –216
C.
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
Example 2: Finding Roots of Fractions
Find the root.
Think: What number squared
equals
A.
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
Additional Example 2: Finding Roots of Fractions
Find the root.
Think: What number cubed equals
B.
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
Additional Example 2: Finding Roots of Fractions
Find the root.
Think: What number squared
equals
C.
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
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.
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
Example 3: Art Application
As part of her art project, Ashley 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.
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.
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
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
The side length of the paper square is
Since 3.6 is too low and 3.65 is too high, is between 3.6 and 3.65. Round to the nearest tenth.
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
The symbol ≈ means “is approximately equal to.”
Writing Math
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
Estimate to the nearest tenth the side length of a cube with a volume of 26 ft3.
Example 4
Since the volume of the cube is 26 ft3, then the length of each side of the cube is ft. 26 is not a perfect cube, so find two consecutive perfect cubes that is between: 8 and 27. is between and , or 2 and 3.
Since 26 and 27 are very close, ≈ 3.0.
The side length of the cube is ≈ 3.0 ft.
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
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, …
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
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).
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
Irrational numbers are 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.
Holt McDougal Algebra 1
1-5 Roots and Real 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.
Holt McDougal Algebra 1
1-5 Roots and Real NumbersThe real numbers are made up of all rational and irrational numbers.
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
Note the symbols for the sets of numbers.R: real numbersQ: rational numbersZ: integersW: whole numbersN: natural numbers
Reading Math
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
Example 1: Classifying Real Numbers
Write all classifications that apply to each real number.
A.
–32 = –
32 1
rational number, integer, terminating decimal
B.
irrational
–32
–32 can be written in the form .
14 is not a perfect square, so is irrational.
–32 can be written as a terminating decimal.
–32 = –32.0
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
Write all classifications that apply to each real number.
a. 7
rational number, repeating decimal
Example 2
67 9 = 7.444… = 7.4
7 can be written in the form .49
can be written as a repeating decimal.
b. –12 –12 can be written in the form .
–12 can be written as a terminating decimal.
rational number, terminating decimal, integer
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
Write all classifications that apply to each real number.
Example 2
irrational
100 is a perfect square, so is rational.
10 is not a perfect square, so is irrational.
10 can be written in the form and as a terminating decimal.
natural, rational, terminating decimal, whole, integer
Holt McDougal Algebra 1
1-5 Roots and Real Numbers
Find each square root.
1. 2. 3. 4.3
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.
Lesson Quiz
Write all classifications that apply to each real number.
6. –3.89 7.rational, repeating decimal
irrational
15