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

Roots and Irrational Numbers Section 1.5

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.

Objectives: In this lesson you’ll: Evaluate expressions containing roots. Classify numbers within the real number system 4

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)

Square Roots Perfect Square Roots 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

Are squares and square roots inverses? Start Square it Root 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

Square Roots Find the square roots of 16. Solution Positive real numbers have two square roots. Find the square roots of 16. Solution 4  4 = 42 = 16 = 4 Positive square root of 16 (–4)(–4) = (–4)2 = 16 – = –4 Negative square root of 16 The square roots of 16 are 4 and - 4.

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 . Cube roots 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.

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

You try Finding Roots of Fractions. Think: What number squared equals Think: What number cubed equals b.

You try Finding Roots of Fractions. Think: What number squared equals Think: What number cubed equals B. = –6 (–6)(–6)(–6) = 36(–6) = –216

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. 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

Approximating Square Roots Approximate to the nearest whole number. Solution Is between 7² and 8².

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

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

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 9 a. 7 can be written as a repeating decimal. 67  9 = 7.444… = 7.4 rational number, repeating decimal b. –12 –12 can be written in the form . –12 can be written as a terminating decimal. rational number, terminating decimal, integer

Write all classifications that apply to each real number. 10 is not a perfect square, so is irrational. irrational 100 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

A challenge… Would you know how to solve this…. -11 -11 x = 5

A challenge… Solve the variable. -3 -3 x = 8

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