Squares and Square Roots  

Vocabulary Square root Perfect square Radical sign Irrational numbers

Guiding Questions How would you define “square root”? When will a square root be a whole number? When will it be irrational? Can a square root be a decimal as 2.25? Explain What connection can be made between the side length of a square and the area of the square? If a square as an area of 100, what is the perimeter?

By definition Ö25 is the number you would multiply times itself to get 25 for an answer. Because we are familiar with multiplication, we know that Ö25 = 5 Numbers like 25, which have whole numbers for their square roots, are called perfect squares You need to memorize at least the first 15 perfect squares

Perfect square Square root Perfect square Square root 1 Ö1 = 1 81 Ö81 = 9 4 Ö4 = 2 100 Ö100 = 10 9 Ö9 = 3 121 Ö121 = 11 16 Ö16 = 4 144 Ö144 = 12 25 Ö25 = 5 169 Ö169 = 13 36 Ö36 = 6 196 Ö196 = 14 49 Ö49 = 7 225 Ö225 = 15 64 Ö64 = 8

Every whole number has a square root Most numbers are not perfect squares, and so their square roots are not whole numbers. Most numbers that are not perfect squares have square roots that are irrational numbers Irrational numbers can be represented by decimals that do not terminate and do not repeat The decimal approximations of whole numbers can be determined using a calculator

Obj: To find the square root of a number Find the square roots of the given numbers If the number is not a perfect square, use a calculator to find the answer correct to the nearest thousandth. 81 Ö81 = 9 37 Ö37 » 6.083 158 Ö158 » 12.570

Obj: To find the square root of a number Find two consecutive whole numbers that the given square root is between Try to do this without using the table Ö16 = 4 and Ö25 = 5 so Ö18 Ö18 is between 4 and 5 Ö115 Ö100 = 10 and Ö121 = 11 so Ö115 is between 10 and 11

Obj: To find the square root of a number Square and square root are inverse operations (they undo each other) Find the length of a square by using inverse operations   Area = 36 units2

A square window has an area of 169 square inches A square window has an area of 169 square inches. How wide is the window?  

A square shaped kitchen table has an area of 16 square feet A square shaped kitchen table has an area of 16 square feet. Will it fit through a van door that has a 5 foot wide opening?  

A square window has an area of 169 square inches A square window has an area of 169 square inches. What is the perimeter of the window?  

Guiding Questions (you should be able to answer) How would you define “square root”? When will a square root be a whole number? When will it be irrational? Can a square root be a decimal as 2.25? Explain What connection can be made between the side length of a square and the area of the square? If a square as an area of 100, what is the perimeter?

Multiplying radicals The product of the square roots of two numbers is the same as the square root of the product of the numbers Examples: Ö3 · Ö12 = Ö36 Ö7 · Ö11 = Ö77

Simplify the following expressions -Ö4 = -2 7Ö64 + 9 = 7 8 + 9 · = 56 + 9 = 65 5 + Ö25 Ö49 = 5 5 + 7 · = 25 + 7 = 32

Ö Ö Ö Simplify the following expressions = = 1 1 – = – 6 12 2 1 = – 12 4 Ö4 2 = = 81 Ö81 9 1 1 Ö 1 Ö 1 – = – 36 6 12 144 2 1 = – 12 12 1 = 12

Simplified radical form No factor inside the radical should be a perfect square. Ö18 = Ö9 2 = Ö9 Ö2 = 3 Ö2 · Ö108 = Ö36 3 = Ö36 Ö3 = 6 Ö3 · Ö96 = Ö16 6 = Ö16 Ö6 = 4 Ö6 ·