3 CaliforniaStandards2.0 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 system4
5 Words to know…Square root - a number which, when multiplied by itself, produces the given number. (Ex. 7² = 49, 7 is the squareroot 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)
7 Are squares and square roots inverses? StartSquare itRoot itResult335599A square root is the inverse operation of a square!
8 Do you know your perfect squares? 7 and -7258 and -81211963 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.Solution4 4 = 42 = 16= 4Positive squareroot of 16(–4)(–4) = (–4)2 = 16–= –4Negative squareroot of 16The square roots of 16 are 4 and - 4.
10 Writing MathThe 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 rootsA 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 equalsb.
13 You try Finding Roots of Fractions. Think: What number squared equals Think: What number cubed equalsB.= –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.SolutionIs 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 3Between 4 and 5Between 4 and 5Between 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 numbersQ: rational numbersZ: integersW: whole numbersN: natural numbersReading Math
20 Classifying Real Numbers Write all classifications that apply to each real number.A.–32321–32 can be written in the form .–32 = ––32 can be written as a terminating decimal.–32 = –32.0rational number, integer, terminating decimalB.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 9a. 7can be written as a repeating decimal.67 9 = 7.444… = 7.4rational number, repeating decimalb. –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, sois irrational.irrational100 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….-11-11x = 5
25 Lesson QuizFind each square root.220.127.116.11.15. 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. –3.89rational, repeating decimal7.irrational