Square Roots and Cubes Roots of Whole Numbers Tammy Wallace

Perfect Squares A number is a perfect square if it is the product of a number and itself. The first 12 perfect squares: NUMBER MULTIPLIED BY ITSELF PERFECT SQUARES 1 x 1 = 2 x 2 = 3 x 3 = 4 x 4 = 5 x 5 = 6 x 6 = NUMBER MULTIPLIED BY ITSELF PERFECT SQUARES 7 x 7 = 8 X 8 = 9 X 9 = 10 x 10 = 11 x 11 = 12 x 12 = 1 49 4 64 9 81 16 100 25 121 36 144

Square Roots Taking the square root of a number is the inverse operation to squaring a number (or multiplying a number by itself). For example if 3 2 = _________, then 9 =_________. The symbol tells you to 9 3 find the number that multiplies by itself to give you the inside number.

Parts of a Radical An expression that contains a square root is called a radical. It can have three parts. Radicand: the value you are taking the ‘root’ of Index: tells you which root you are looking for; if there is not index, it is assumed to be 2. Coefficient: it is being multiplied by the radical.

Simplify 100 = 25 = 144 = 𝟏𝟎 𝟓 𝟏𝟐

Non-Perfect Radicals Simplify: 24 Since 24 is not a perfect square, its square root is not a whole number. To simplify this radical, 24 needs to be rewritten as the product of two factors. However, one of the factors must be a perfect square.

Simplify 24 What is the highest factor of 24 that is also is a perfect square? 4 Therefore, 24 = ____ X _____ 24 = 4∙6 = 4 ∙ 6 =𝟐 𝟔 4 6

Simplify 32 What is the highest factor of 32 that is also is a perfect square? 16 Therefore, 32 = ____ X _____ 32 = 16∙2 = 16 ∙ 2 =𝟒 𝟐 16 2

Simplify 54 What is the highest factor of 54 that is also is a perfect square? 9 Therefore, 54 = ____ X _____ 54 = 9∙6 = 9 ∙ 6 =𝟑 𝟔 9 6

Perfect Cubes A number is a perfect cube if it is the product of a number and itself three times. The first 5 perfect cubes: NUMBER MULTIPLIED BY ITSELF 3 TIMES PERFECT CUBES 1 x 1 x 1 = 2 x 2 x 2 = 3 x 3 x 3 = 4 x 4 x 4 = 5 x 5 x 5 = 1 8 27 64 125

Cube Roots Taking the cube root of a number is the inverse operation of cubing a number. For example if 2 3 = _________, then 3 8 =_________. So, the symbol 3 tells you to find the number that multiplies by itself three times to give you the inside number – the radicand. 8 2

Simplifying the following Radical Expressions 3 125 = 3 27 = 3 64= 5 3 𝟒

Non-Perfect Cubes =3 3 2 Simplify 3 54 2 27 What is the highest factor of 3 54 that is also is a perfect cube? 𝟐𝟕 Therefore, 54 = ____ X _____ 3 54 = 3 27∙2 = 3 27 ∙ 3 2 =3 3 2 27 2

Simplify 3 24 What is the highest factor of 24 that is also is a perfect cube? 𝟖 Therefore, 54 = ____ X _____ 3 24 = 3 8∙3 = 3 8 ∙ 3 3 =2 3 3 𝟖 𝟑

Simplify 3 128 What is the highest factor of 128 that is also is a perfect cube? 64 Therefore, 54 = ____ X _____ 3 128 = 3 64∙2 = 3 64 ∙ 3 2 =4 3 2 𝟔𝟒 𝟐