Operations with Square Roots # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 # squared 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 All of the work with square roots becomes soooooooooooooo much easier if you memorize this list of the first 20 perfect squares.

There are two numbers that if you square them you will get 100. What number times itself is equal to 100? There are two numbers that if you square them you will get 100. 10 x 10 = 100 and -10 x -10 = +100

There are two answers when you take a the square root of a number if you are simply asking “What number times itself equals___________”. There will be a positive number and a negative number If you use a calculator, it will only give you the positive number. Symbolically textbooks try to let you know which answer they are looking for. They will have symbols to show you if you should give the positive root, negative root, or both. 100 means they want the positive answer because they didn’t write anything in front of the radical. − 81 means they want the negative answer because they wrote a negative sign in front of the radical. ± 64 means they want you to give both the positive and negative answers because they put a “plus or minus” sign in front of the radical

What is the value of 1 ? What is the value of - 9 ? 1 = 1 - 9 = - 3  16 =  4 25 = 5 - 50  -7.1 There is no number that you can square and get a negative number

Square roots of fractions Before you try to convert your fraction to a decimal, look at the numbers. Are the numerator and denominator made of perfect squares? If they are, you just need to take the square root of the numerator and take the square root of the denominator. 64 225 = 64 225 = 8 15 That is much easier than trying to use the decimal and find 0.2844444444444444

The same rules apply about the sign in front of the radical as to whether you should give the positive or negative answer when dealing with fractions. 169 196 = 169 196 = 13 14 − 100 225 =− 100 225 = - 10 15 =− 2 3 ± 9 16 =± 9 16 = ± 3 4

What if you had a decimal that that didn’t appear to be a perfect square or a number that was larger than your memorized list of perfect squares? You could use a factor tree with the prime factorization to help you find the square root. 20.25 = ? As an estimate you know that the number should be between 4 and 5 since 4 squared is 16 and 5 squared is 25 Ignore the decimal point for a moment and think about 2025. Can you find factor pairs of 2025? You can rearrange these numbers in pairs so that they are with a different number 5 x 5 x 9 x 9 can be written as 5 x 9 x 5 x 9 which is equal to x 45 That means that 45 squared is 2,025 which would mean that 4.5 squared is 20.25 I know that if I had a bunch of change in my pocket I could make 20.25 using just quarters. How many would it take? 2025 25 x 81 5 x 5 9 x 9

Let’s try another one for a number that is much larger than a number on our perfect square list 4624 = ?

Let’s try another one for a number that is much larger than a number on our perfect square list 4624 = ? Since you were finding the square root of 4624 you were looking for one number times itself to get 4624. You need to take your list of prime factors and split them into two groups with same factors 2x2x2x2x17x17 4624 4 x 1156 2 x 2 2 x 578 2 x 289 17 x 17 2x2x17 2x2x17 = 4624 x 68 68 = 4624 x The prime factorization is 2x2x2x2x17x17 or 2^4 x 17^2 4624 = 68

Using the Order of Operations and Square Roots Parentheses , Exponents, Multiplication or Division from left to right, Addition or Subtraction from left to right 36 +12−4∙5 6+12−4∙5 6 + 12 - 20 18-20 -2 4 225 ÷5−4∙2 4(15)÷5−4∙2 60 ÷5−4∙2 12−4∙2 12−8 4 Keep in mind that radicals count in the EXPONENT category

Applications to geometry problems If you know the area of a square you will be asked to find the side length of that square. You need to keep in mind that to get the area of a square you multiply the length times the width, but they happen to be the same number so you are squaring the side. Find the side length for a square with an area of 40 sq cm. As an estimate you know that the number should be between 6 and 7 since 6 squared is 36 and 7 squared is 49. 40 is 4 away from 36, but 9 away from 49, so it is about one third of the way between those values. 6.3 is a good estimate. Find s if 𝑠 2 = 84. As an estimate you know that the number should be between 9 and 10 since 9 squared is 81 and 10 squared is 100. 84 is 3 away from 81, but 16 away from 100, so it is about one sixth of the way between those values. 9.1 or 9.2 is a good estimate.

Applications to geometry problems 𝐴=𝜋 𝑟 2 is the formula for area of a circle Normally when you find the area of the circle you square the radius and then multiply that by 3.14. In these problems you are not multiplying by 3.14, but you are keeping the Pi symbol in the problem. So the value in front of the π symbol is telling you the value of the radius squared. You are looking for just the radius. Find the radius if the area of the circle is 64𝜋 square units. Here you are just trying to figure out what the radius would be if the radius squared is 64. the radius would have to be 8 since 8 x 8 = 64.