Squares, Square Roots, Cube Roots, & Rational vs. Irrational Numbers

Perfect Squares Can be represented by arranging objects in a square.

Perfect Squares

1 x 1 = 1 2 x 2 = 4 3 x 3 = 9 4 x 4 = 16 Activity: Calculate the perfect squares up to 15 2 … Perfect Squares

1 x 1 = 1 2 x 2 = 4 3 x 3 = 9 4 x 4 = 16 5 x 5 = 25 6 x 6 = 36 7 x 7 = 49 8 x 8 = 64  9 x 9 = 81  10 x 10 = 100  11 x 11 = 121  12 x 12 = 144  13 x 13 = 169  14 x 14 = 196  15 x 15 = 225 Perfect Squares

Square Numbers One property of a perfect square is that it can be represented by a square array. Each small square in the array shown has a side length of 1cm. The large square has a side length of 4 cm. 4cm 16 cm 2

The large square has an area of 4cm x 4cm = 16 cm 2. The number 4 is called the square root of 16. We write: 4 = 16 4cm 16 cm 2 Square Numbers

The opposite of squaring a number is taking the square root. This is read “the square root of 81” and is asking “what number can be multiplied by itself and equal 81?” 9 X 9 = 81so The square root of 81 is 9

Is there another solution to this problem? 9 X 9 = 81 Yes!!! So… 9 & -9 are square roots of X -9 = 81 as well!

Simplify Each Square Root

Simplify Each Square Root 8 - 7

What About Fractions? = Take the square root of numerator and the square root of the denominator

What About Fractions? So…the square root of is…………

What About Fractions? = Take the square root of numerator and the square root of the denominator

What About Fractions? So…the square root of is…………

Think About It Do you see that squares and square roots are inverses (opposites) of each other?

Estimating Square Roots Not all square roots will end-up with perfect whole numbers When this happens, we use the two closest perfect squares that the number falls between and get an estimate

Estimating Square Roots Example: Estimate the value of each expression to the nearest integer. Is not a perfect square but it does fall between two perfect squares. 25 and 36

Estimating Square Roots 56 Since 28 is closer to 25 than it is to 36, ≈ 5

Estimating Square Roots Example: Estimate the value of each expression to the nearest integer. Is not a perfect square but it does fall between two perfect squares. 36 and 49

Estimating Square Roots 67 Since 45 is closer to 49 than it is to 36, ≈ 7

Estimating Square Roots Example: Estimate the value of each expression to the nearest integer. Is not a perfect square but it does fall between two perfect squares and -121

Estimating Square Roots Since -105 is closer to -100 than it is to -121, ≈ -10

Estimating Square Roots Practice: Estimate the value of the expression to the nearest integer. ≈ - 5 ≈ 7

Rational vs. Irrational Real Numbers – include all rational and irrational numbers Rational Numbers – include all integers, fractions, repeating, terminating decimals, and perfect squares Irrational Numbers – include non-perfect square roots, non-terminating decimals, and non-repeating decimals

Rational vs. Irrational Examples: Rational; the decimal repeats Irrational; not a perfect square Rational; is a fraction Irrational; decimal does not terminate or repeat

Rational vs. Irrational Practice: Irrational; Pi is a decimal that does not terminate or repeat Irrational; not a perfect square Rational; is a perfect square Rational; the decimal terminates π

Cube Roots To “Cube” a number we multiply it by itself three times =4 x 4 x = 64

Cube Roots Remember that taking the “cube root” of a number is the opposite of cubing a number. =5 x 5 x 5 5 is the cube root of 125

Cube Roots Remember that taking the “cube root” of a number is the opposite of cubing a number. =-3 x -3 x is the cube root of - 27

Simply Each Cube Root

Simply Each Cube Root 9 - 2