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

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

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Perfect Squares

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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

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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

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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

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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

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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

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

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Simplify Each Square Root 10 - 4

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Simplify Each Square Root 8 - 7

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

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

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

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

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

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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

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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

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

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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

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

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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. -100 and -121

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Estimating Square Roots -10-11 Since -105 is closer to -100 than it is to -121, ≈ -10

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

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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

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Rational vs. Irrational Examples: - 0.81 Rational; the decimal repeats Irrational; not a perfect square Rational; is a fraction 0.767667666... Irrational; decimal does not terminate or repeat

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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 - 0.456 π

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Cube Roots To “Cube” a number we multiply it by itself three times =4 x 4 x 4 4 3 4 3 = 64

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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

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

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Simply Each Cube Root 10 - 6

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Simply Each Cube Root 9 - 2

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