Alg. 1 Radical Notes

Alg. 1 Radical Notes Evaluating Square Roots and Cube Roots (Day 1) Objective: SWBAT find the square root and cube roots of monomials Perfect Squares: Perfect Cubes: 1 =11 1 = 11 =1111 11 1 =111 1 1 = = 9 1 =11 1 = 8 = 9 8 =88 1 =11 19 = 8 888 1 = 1 9 = 9 9 81 1 =11 19 = 9 9 9 9 9 = 10 =1010 100 1 =11 = 1 10 10 10 10 1000 Roots~ the same number that multiplies to make another Factor Tree~ a math technique to find the roots What leaves? Pairs What stays in? Singles Evaluate~ your answer is a number Simplify~ putting into simplest terms

Evaluate the following radicals. 1). ) ) 1x ) 00y 8 8 8 Because the negative is inside the root it is imaginary 1 1 1 1x x x x 00 /\ 0 0 y 0y y y y y y i Cube Root~ Looking for cubes to leave. Pairs and singles stay inside. No plus or minus. 8. 8x 9. 10. x 9 / \ 8 x / \ / \ xxx / \ x / \ / \ 9 x x x x x No imaginary numbers with Cube Roots

Estimating Square Roots 1. Pick a perfect square above the number, and a perfect square below the number (see chart on first page).. Estimate / Round to the nearest perfect square. Estimate the following square roots. 18 Perfect Square Below Perfect Square Above Estimation 1 18 is in-between and. Since it is closer to estimate it is. Perfect Square Below Perfect Square Above Estimation 9 is in-between and. Since it is closer to estimate it is. Perfect Square Below Perfect Square Above Estimation 8 81 9 is in-between 8 and 9. Since it is closer to 9 estimate it is 8. Perfect Cube Below Perfect Cube Above Estimation is in-between and. Since it is closer to estimate it is.

Perfect Cube Below Perfect Cube Above Estimation 1 1 1 8 is in-between 1 and. Since it is closer to 1 estimate it is 1. Perfect Cube Below Perfect Cube Above Estimation 8 Evaluate the following expression if x = Plug the value into x, and perform PEMDAS 1 is in-between - and -. Since it is closer to - estimate it is -. ) x ) x x 8 8 8 8 / \ 0 8

Homefun: Must be done on a different sheet of Paper Evaluate the following square roots and cube roots. 1) 11 ) ) ) x ) ) 81 ) 9 8) 8 9) x 10) 1 11) 1 1) 100y 1) 1) 1 1) 1y Approximate the value of the radical by listing the two integers that the radical lies between. 1) 1) 1 18) 88 19) 0 Approximate the radical to the nearest integer. 0) 1 1) 10 ) 0 ) Order the following radicals from least to greatest. ) 81,,, 8, 9, 100 Evaluate the following expression if x = 9 Evaluate the following expression if x = 8 ) x ) x ) x x

Simplifying Radical Expressions (Day ) Objective: SWBAT Simplify square roots and cube roots Simplify the following radicals. 1) 1 ) 18 Factor Tree Leaves Stays In Factor Tree Leaves Stays In 1 18 PAIRS SINGLES 9 SINGLES PAIRS ) 0x ) 8 Factor Tree Leaves Stays In Factor Tree Leaves Stays In 0 x / \ x x x PAIRS x x SINGLES x 8 1 SINGLES PAIRS x x

) ) y Factor Tree Leaves Stays In Factor Tree Leaves Stays In 9 PAIRS SINGLES y y / \ y y y y y PAIRS y y y y y y y y y SINGLES y y ) a 8) 81 9 9 / \ a a a a a a a a a All Singles 9 18 : : 18a a

9) 10) 0 Factor Tree Leaves Stays In Factor Tree Leaves Stays In 9 TRIOS SINGLES or PAIRS 0 8 TRIOS SINGLES or PAIRS 11) x 1) 19 8 9 / \ xxx x x x x x 19 1 1 9 : x 9x

Homefun: Must be done on a different sheet of Paper Simplify the following square roots and cube roots. 1) ) 98 ) 9x ) 00 ) ) ) 8 a 8) 0 9) 108 10) 00 11) 1 1) 81y 1) 19 1) 00 1) 80

Multiplying square roots Multiplying Radicals Objective: SWBAT multiple and divide square roots and cube roots 1. Multiply the insides together, then factor tree. Multiply all outsides together. Write as one radical answer 1) 1 Inside Outside 1 No Other Outsides ) 0 Inside Outside 0 100 10 10 No Other Outsides 10

) x 18x 18 108 108 18 9 Inside Outside x x x x x x x No Other x Outsides ) Multiply Only the Insides Inside Outside 18 Multiply Write ALL 18 Outside Outsides And ( Including from Inside 9 the Root) Together 18 18 ) y y 1y Inside Outside Multiply Only the Insides 8 8 y y y y y y y Multiply ALL Outsides ( Including the Root) from y y 8y Write Outside And Inside Together 8y

) ( ) ( ) ( ) Inside Outside Multiply Only the Insides 9 9 Multiply ALL Outsides ( Including from the Root) 8 Since there are no insides 8 8) 1y 8y 9) 1 10) 10 100 1y 8y y 1 8 10 100 1000 y 8 1 1 y y y y y y y 8 1 1000 100 10 10 10 : 1 y 10 : 10y :

11) 1a a 1) 1 1 a a a 108 108 18 9 a : : a

Homefun: Must be done on a different sheet of Paper Simplify the following radicals. 1) 10 ) 1 ) 0 0 ) 18x 8x ) 8 ) ) 8) 11) 1x 1) 8 1) 9 y 9) 10) 18 1) 10 1 1) 18

Review: Adding and Subtraction Radical Expressions Objective: SWBAT add and subtract square roots and cube roots 1) x + + x 9 ) 8x + y + x y x x9 8x x yy x 1 x y You can only combine Like terms. Adding and Subtracting Square Roots 1. You can only combine like roots. The root never changes only the coefficients when you add/subtract radicals.. Do a factor tree for each term. Add / subtract coefficients. Simplify the following. 1) ) 8 ) Like Radicals Like Radicals Like Radicals except 8 1 1

) 0 Unlike Roots ) 00 98 Unlike Roots 1 st Term Factor Tree nd Term Factor Tree 1 st Term Factor Tree nd Term Factor Tree 0 1 00 100 10 10 98 9 10 Rewrite Simplified Expression Add / Subtract Coefficients Rewrite Simplified Expression Add / Subtract Coefficients 10 10 9 ) 81 Unlike Roots ) 1 st Term Factor Tree nd Term Factor Tree 0 Unlike Roots 8 / \ 81 / \ 0 1 / \ / \ Rewrite Simplified Expression Add / Subtract Coefficients

Remember to combine like terms Remember of your Order of Operations ) 1 8) 1 1 1 ( ) 18 9) 9 1 18 10) 9( 1 18) 108 1 Unlike Roots Can ' t Combine Unlike Roots Can ' t Combine Homefun: Must be done on a different sheet of Paper Simplify the following radicals. 1) ) ) 8 ) ) 0 18 ) 1 ) 8) 10 9) 10) 11) 8 9 1 1) 9 1 1) 8 18 1) 1) 0 100 80

Dividing Radical Expressions Objective: SWBAT divide square roots and cube roots Division Property of Radicals: a a b b You can reduce the fraction first, then take the square root or You can take the square root first then reduce. Simplify the following monomials 1) 9 9 ) 18 18 81 81 9 9 /9 ) /9 1 1 ) 1 1 1 ) 1 1 1 ) 18 18 1 1 8 8 1

Rationalizing the denominator 1. Can t have a radical in the denominator. Reduce your radicals first always. Multiply the numerator and denominator by the radical and simplify ) Can you Reduce the Insides? No Re ducing Rationalize the Denominator Factor Tree /Reduce Outsides? No More Simplifying : 8) 1 18 Can you Reduce the Insides? 1 1/ 18 18 / Rationalize the Denominator 9 Factor Tree /Reduce Outsides? No More Simplifying :

9) Can you Reduce the Insides? Rationalize the Denominator 9 Factor Tree /Reduce Outsides? Can' t Simplify and : 10) Can you Reduce the Insides? Rationalize the Denominator Factor Tree /Reduce Outsides? No Re ducing Because it is a Cube Root We Need A Trio instead of Pairs 8 No More Simplifying :

11) Can you Reduce the Insides? Rationalize the Denominator Factor Tree /Reduce Outsides? Because it is / 1 1 / a Cube Root We Need A Trio instead of Pairs 1 1 No More Simplifying : 1) 9 9 Reduce Outsides 9

Homefun: Must be done on a different sheet of Paper Simplify the following radicals. 1) 11 19 ) ) 1 ) 8 ) 1 ) ) 8) 11 9) 1 10) 11) 1 1) 1) 1) 10 1

Dividing Radical Expressions (Day 9) Objective: SWBAT divide square roots and cube roots 1) 8 Can you Reduce the Insides? 8 1 / / Rationalize the Denominator 9 Factor Tree /Reduce Outsides? No More Simplifying : ) Can you Reduce the Insides? Rationalize the Denominator Factor Tree /Reduce Outsides? /18 /18 9 Can' t Simplify and :

) Can you Reduce the Insides? Simplify 8 Rationalize the Denominator 8 8 8 1 Factor Tree /Reduce Outsides? No More Simplifying : 1 ) 9 Can you Reduce the Insides? Rationalize the Denominator Factor Tree /Reduce Outsides? 9 1 Re duce Outsides 1 Don 't Need To Rationalize

) 18 /18 /18 18 1 1 9 9 9 1 ) 0 9 Homefun: Must be done on a different sheet of Paper Simplify the following radicals. 1) 8 ) ) 9 ) 0 ) 1 8 ) ) 1 8 8) 0 9) 1 10) 11) 10 1) 8 9 1) 1 1) 18