________ Find the sides of the following squares given the area of each. Hints: How is the area of a square found? multiply the sides What is true about the sides of a square? they are equal Therefore the sides must be the same and multiply to = the area.

By finding the sides of the squares you found the____ ___ of the area. 5 20 2x 3x3 By finding the sides of the squares you found the____ ___ of the area. A = 400 A = 9x6 A = 4x2 A= 25

Square Roots 5 20 2x 3x3 By finding the sides of the squares you found the square roots of the areas. Square root - A = 400 A = 9x6 A = 4x2 A= 25

Square Roots A = 400 5 20 2x 3x3 By finding the sides of the squares you found the square roots of the areas. Square root - the side of a square - a #/term that multiplies by itself to equal a certain #/term A = 9x6 A= 25 A = 4x2

Square Roots What are the square roots of 16? 4 since (4)(4)=16 and -4 since (-4)(-4)=16 Therefore any positive # has 2 square roots, a positive and a negative. The positive root is known as the principal root.

Square Roots √ asks for the positive root. Ex. √36 = 6 -√ asks for the negative root. Ex. -√64 = -8 +√ asks for both roots. Ex. + √81 = + 9 Evaluate. a) √49 b) -√100 c)+√25x2 d) √-36

Square Roots Evaluate. √49 b) -√100 c)+√25x2 d) √-36 = 7 = -10 = + 5x = n.p A negative # can not have a square root since no # times itself = a neg.

Square Roots The √ symbol is called a radical. The term under the radical is called the radicand. Ex. √25 radical radicand

Square Roots Numbers such as 1, 4, 9, 16, 25, 36….. are known as _______ ________

Square Roots Numbers such as 1, 4, 9, 16, 25, 36….. are known as square #s or perfect squares. Square # - a # that has a whole # as a square root

Square Root Investigation # 1 Evaluate the following. = 4 2.a) √10 =3.16 3.a) √256 =16 4.a) √81 =9 5.a) √12 =3.46 b) √4 √4 =2∙2 =4 b) √2 √5 =1.41∙2.24 =3.16 b) √16 √16 =4∙4 =16 b) √9 √9 = 3∙3 = 9 b) √4 √3 =2∙1.73 =3.46

What is true about (a) and (b) in each? equal What would be true about √60 and √20 √3 ? How else could you write √50 ? √28 ? ex. √50 = √25 √2 √28 = √4 √7

√ Conclusion # 1 The √ of a # is equal to the √ ’ s of its factors multiplied (or vice versa). Ex. √20 √27 √99 = √4 √5 = √9 √3 = √9 √11 √3 √7 √4 √9 √ 16 √5 =√21 = √36 = √80

Use the above conclusion to simplify the following Use the above conclusion to simplify the following. Write the expression in an equivalent, but different way. Don’t evaluate and express as a decimal approximation. (Hint: think of the conclusion just made and express the √ in a different way) a) √8 = √4 √2 = 2√2 2√2 is considered ‘simpler’ than √8 since the radicand in 2√2 is smaller than in √8

b) √27 = √9 √3 = 3 √3 c) √150 = √25 √6 = 5 √6

3 has no perfect square factors (other than 1) The square root of a non-perfect square is in simplest form/simplest radical form when the radicand does not have any factors that are perfect squares (other than1). Ex. √12 = √4 √3 =2 √3 3 has no perfect square factors (other than 1) √6 already in simplest form since 6 has no perfect square factors

To simplify a radical: -factor the radicand (try to factor so that you get a perfect square factor) ex. √20 = √4 √5 -find the roots of the perfect square factors and leave the non-perfect squares as √ √20 = 2 √5 -continue until the √ is in simplest form (radicand does not contain any factors that are perfect squares) 2 √5 5 does not contain any factors that are perfect squares

Simplify. a) √32 b) √300 c) √99 d) √10

√32 = √16 √2 = 4 √2 b) √300 = √25 √12 = 5 √12 = 5 √4 √3 = 5 ∙ 2 √3 = 10 √3 c) √99 = √9 √11 = 3 √11 d) √10 can not be simplified further since 10 has no factors that are perfect squares

√ Investigation # 2 Evaluate the following. =2(3) + 3(3) =6 + 9 =15 2.a) 5 √16 + 6 √16 =5(4) + 6(4) =20 + 24 =44 3.a) -4 √25 + 2 √25 = -4(5) + 2(5) = -20 + 10 = -10 4.a) 7 √4 - 3 √4 =7(2) – 3(2) =14 – 6 = 8 5.a) 10 √100 - 9 √100 = 10(10) – 9(10) = 100 – 90 = 10 6.a) 2 √4 + 3 √16 = 2(2) + 3(4) = 4 + 12 = 16 b) 5 √9 =5(3) =15 b) 11 √16 =11(4) =44 b) -2 √25 = -2(5) = -10 b) 4 √4 = 4(2) = 8 b) 1 √100 =1(10) = 10 b) 5 √20 =5(4..5) =22.5

What is true about the (a) and (b) in each? equal, except # 6 where the radicands were different What would be true about 5 √25 + 3 √25 and 8 √25 ? equal What would be true about 12 √4 - 7 √4 and 5 √4 ? Could 3 √10 + 10 √10 be expressed in a simpler way ? 13 √10 Could 9 √7 - 15 √7 be simplified ? -6 √7

√ Conclusion # 2 An expression consisting of a sum or difference of radicals with the same radicands can be simplified by adding or subtracting the numbers in front of the radicals/the coefficients and keeping the radicand the same. Ex. a) 8 √3 + 4 √3 b) 2 √4 + 3 √4 c) 10 √25 - 7 √25 =12 √3 = 5 √4 = 3 √25 = 5 (2) = 3(5) = 10 = 15 2 √3 + 3 √5 not possible since the radicands are not the same

Simplify the individual radicals first = 5 √9 √5 - 3 √4 √5 can’t be simplified as is because the radicands are different, but √8 can be simplified = 2 √2 + 3 √4 √2 = 2 √2 + 3 (2) √2 = 2 √2 + 6 √2 = 8 √2 f) 5 √45 - 3 √20 Simplify the individual radicals first = 5 √9 √5 - 3 √4 √5 = 5(3) √5 – 3(2) √5 = 15 √5 - 6 √5 = 9 √5 g) 6 √99 + 10 √44 = 6 √9 √11 + 10 √4 √11 = 6(3) √11 + 10(2) √11 = 18 √11 + 20 √11 = 38 √11