6 A number that is multiplied by itself to form a product is called a square root of that product.The operations of squaring and finding a squareroot are inverse operations.The radical symbol , is used to represent square roots. Positive real numbers have twosquare roots.= 4Positive squareroot of 164 4 = 42 = 16(–4)(–4) = (–4)2 = 16–= –4Negative squareroot of 16
7 The nonnegative square root is represented by The nonnegative square root is represented by The negative square root is represented by – .A perfect square is a number whose positive square root is a whole number. Some examples of perfect squares are shown in the table.14916253649648110002122232425262728292102
8 The expression does not represent a real number because there is no real number that can be multiplied by itself to form a product of –36.Reading Math
9 Example 1: Finding Square Roots of Perfect SquaresFind each square root.A.Think: What number squared equals 16?42 = 16Positive square root positive 4.= 4B.Think: What is the opposite of thesquare root of 9?32 = 9= –3Negative square root negative 3.
10 Example 1C: Finding Square Roots of Perfect SquaresFind the square root.Think: What number squared equals ?2581Positive square root positive .59
11 Check It Out! Example 1Find the square root.1a.22 = 4Think: What number squaredequals 4?= 2Positive square root positive 2.1b.52 = 25Think: What is the opposite of the square root of 25?Negative square root negative 5.
12 The square roots of many numbers like , are not whole numbers The square roots of many numbers like , 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.
13 Example 2: Problem-Solving Application As part of her art project, Shonda willneed to make a square covered in glitter.Her tube of glitter covers 13 squareinches. What is the greatest side lengthShonda’s square can have?Understand the problem1The answer will be the side length of thesquare.List the important information:• The tube of glitter can cover an area of 13 square inches.
14 Example 2 Continued2Make a PlanThe side length of the square is because13.Because 13 is not a perfectsquare, is not a whole number. Estimateto the nearest tenth.=Find the two whole numbers that isbetween. Because 13 is between the perfectsquares 0 and is between and, or between 3 and 4.
15 Because 13 is closer to 16 than to 9, is closer to 4 than to 3. Example 2 ContinuedBecause 13 is closer to 16 than to 9,is closer to 4 than to 3.34You can use a guess-and-checkmethod to estimate
16 Example 2 Continued Solve 3 Guess 3.6: 3.62 = 12.96 too low is greater than 3.6.Guess 3.7: = too highis less than 18.104.22.168.74Because 13 is closer to than to 13.69, is closer to 3.6 than to 3.7. 3.6
17 Example 2 ContinuedLook Back4A square with a side length of 3.6 incheswould have an area of square inches.Because is close to 13, 3.6 inchesis a reasonable estimate.
18 Check It Out! Example 2What if…? Nancy decides to buy more wildflower seeds and now has enough to cover 38 ft2. What is the side length of a square garden with an area of 38 ft2?Use a guess and check method to estimateGuess = too lowis greater than 6.1.Guess = too highis less than 6.2.A square garden with a side length of 6.2 ft would have an area of ft ft is close to 38, so 6.2 is a reasonable answer.
19 All numbers that can be represented on a number line are called real numbers and can be classified according to their characteristics.
20 Natural numbers are the counting numbers: 1, 2, 3, … Whole numbers are the natural numbers and zero:0, 1, 2, 3, …Integers are whole numbers and their opposites: –3, –2, –1, 0, 1, 2, 3, …Rational numbers can be expressed in the form ,where a and b are both integers and b ≠ 0: , , .ab127910
21 Terminating decimals are rational numbers in decimal form that have a finite number of digits:1.5, 2.75, 4.0Repeating decimals are rational numbers indecimal form that have a block of one or moredigits that repeat continuously: 1.3, 0.6, 2.14Irrational numbers cannot be expressed in the form . They include square roots of whole numbers that are not perfect squares and nonterminating decimals that do not repeat: ,, ab
22 Example 3: Classifying Real Numbers Write all classifications that apply to eachReal number.A. –3232 can be written as a fraction and a decimal.321–32 = – = –32.0rational number, integer, terminating decimalB. 55 can be written as a fraction and a decimal.515 = = 5.0rational number, integer, whole number, naturalnumber, terminating decimal
23 Write all classifications that apply to each real number. Check It Out! Example 3Write all classifications that apply to each real number.7 can be written as a repeating decimal.493a. 74967 9 = 7.444… = 7.4rational number, repeating decimal3b. –1232 can be written as a fraction and a decimal.–12 = – = –12.012 1rational number, terminating decimal, integer3c.The digits continue with no pattern.= …irrational number
24 Lesson QuizFind each square root.3712-22.214.171.124.–5. The area of a square piece of cloth is 68 in2.How long is each side of the piece of cloth?Round your answer to the nearest tenth of aninch.8.2 in.Write all classifications that apply to each real number.rational, integer, whole number, natural number, terminating decimal6. 17. –3.89rational, repeating decimal8.irrational