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Roots and Real Number System. January 7, 2013 Mr. Pearson Accelerated Class. Objectives:. In this Review lesson we will: Evaluate algebraic expressions containing roots. Classify numbers within the real number system. Words to know….
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Roots and Real Number System January 7, 2013 Mr. Pearson Accelerated Class
Objectives: In this Review lesson we will: Evaluate algebraic expressions containing roots. Classify numbers within the real number system.
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)
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 Square Roots First semester, I said you should have memorized through what number?
Are squares and square roots inverses? Start Root it Square it Result 3 3 5 5 9 9 A square root is the inverse operation of a square!
Do you know your perfect squares? 7 and -7 25 8 and -8 121 196 3 and -3 Are perfect squares only positive numbers?
Square Roots Positive real numbers have two square roots. Find the square roots of 16. Solution Positive square root of 16 =4 4 4 = 42= 16 = –4 (–4)(–4) = (–4)2= 16 Negative square root of 16 The square roots of 16 are 4 and - 4.
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 . 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. Writing Math Cube roots
You try Find each root. Think: What number squared equals 81? Think: What number squared equals 25? C. Think: What number cubed equals –216? We know that no number times itself can be negative, but what about three numbers? (–6)(–6)(–6) = 36(–6) = –216 = –6 (–6)(–6)(–6) = 36(–6) = –216 = –6
Think: What number squared equals Think: What number cubed equals You try Finding Roots of Fractions. a. b.
Think: What number squared equals Think: What number cubed equals You try… A. B. Finding Roots of Fractions. (–6)(–6)(–6) = 36(–6) = –216 = –6
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 3.872983346... Without a calculator, you can use square roots of perfect squares to help estimate the square roots of other numbers. Remember 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.
Approximating Square Roots Approximate to the nearest whole number. Solution Is between 7² and 8². Is 54 closer to 49 or to 64? Does that place its value closer to 7 or to 8?
Let’s practice… Determine what two consecutive integers each root lies between. Then determine which number it is closer to. Between 2 and 3 Between 4 and 5 Between 4 and 5 Between 5 and 6
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).
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=1.333333333) • 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.
The real numbers are made up of all rational and irrational numbers. Reading Math Note the symbols for the sets of numbers. R: real numbers Q: rational numbers Z: integers W: whole numbers N: natural numbers
–32 can be written in the form . 14 is not a perfect square, so is irrational. 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
7 can be written in the form . 67 9 = 7.444… = 7.4 4 9 can be written as a repeating decimal. –12 can be written in the form . Check It Out! 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
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. Write all classifications that apply to each real number. irrational natural, rational, terminating decimal, whole, integer
A challenge… • Would you know how to solve this…. -11 -11 x = 5 or -5
A challenge… • Solve the variable. -3 -3 x = 8 or -8
