Numerical Roots and Radicals Table of Contents Squares, Square Roots & Perfect Squares Square Roots of Numbers Greater than 400 Estimating Square Roots click on topic to go to that section 1
Squares, Square Roots and Perfect Squares Return to Table of Contents Area of a Square The area of a figure is the number of square units needed to cover the figure. The area of the square below is 16 square units because 16 square units are needed to COVER the figure... 2
Area of a Square The area (A) of a square can be found by squaring its side length, as shown below: A = s 2 A = s 2 A = 4 2 = 4 4 = 16 sq.units Click to see if the answer found with the Area formula is correct! 4 units The area (A) of a square is labeled as square units, or units 2, because you cover the figure with squares... 1 What is the area of a square with sides of 5 inches? 3
2 What is the area of a square with sides of 6 inches? 3 If a square has an area of 9 sq. ft, what is the length of a side? 4
When you square a number you multiply it by itself. 5 2 = 5 5 = 25 so the square of 5 is 25. You can indicate squaring a number with an exponent of 2, by asking for the square of a number, or by asking for a number squared. What is the square of seven? What is nine squared? Make a list of the numbers 1 15 and then square each of them. Your paper should be set up as follows: Number Square 1 1 2 4 3 (and so on) 5
Number Square 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225 The numbers in the right column are squares of the numbers in the left column. If you want to "undo" squaring a number, you must take the square root of the number. So, the numbers in the left column are the square roots of the numbers in the right column. Square Root Square 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225 The square root of a number is found by undoing the squaring. The symbol for square root is called a radical sign and it looks like this: Using our list, to find the square root of a number, you find the number in the right hand column and look to the left. So, the 81 = 9 What is 169? 6
Square Perfect Root Square 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225 When the square root of a number is a whole number, the number is called a perfect square. Since all of the numbers in the right hand column have whole numbers for their square roots, this is a list of the first 15 perfect squares. Find the following. You may refer to your chart if you need to. 7
4 What is 1? 5 What is 81? 8
6 What is the square of 15? 7 What is 256 9
8 What is 13 2? 9 What is 196? 10
10 What is the square of 18? 11 What is 11 squared? 11
12 What is 20 squared? 13 What is the square root of 400? 12
Square Roots of Numbers Greater than 400 Return to Table of Contents Think about this... What about larger numbers? How do you find? 13
It helps to know the squares of larger numbers such as the multiples of tens. 10 2 = 100 20 2 = 400 30 2 = 900 40 2 = 1600 50 2 = 2500 60 2 = 3600 70 2 = 4900 80 2 = 6400 90 2 = 8100 100 2 = 10000 What pattern do you notice? For larger numbers, determine between which two multiples of ten the number lies. 10 2 = 100 1 2 = 1 20 2 = 400 2 2 = 4 30 2 = 900 3 2 = 9 40 2 = 1600 4 2 = 16 50 2 = 2500 5 2 = 25 60 2 = 3600 6 2 = 36 70 2 = 4900 7 2 = 49 80 2 = 6400 8 2 = 64 90 2 = 8100 9 2 = 81 100 2 = 10000 10 2 = 100 Next, look at the ones digit to determine the ones digit of your square root. 14
Examples: Lies between 2500 & 3600 (50 and 60) Ends in nine so square root ends in 3 or 7 Try 53 then 57 53 2 = 2809 List of Squares Lies between 6400 and 8100 (80 and 90) Ends in 4 so square root ends in 2 or 8 Try 82 then 88 82 2 = 6724 NO! 88 2 = 7744 14 Find. List of Squares 15
15 Find. List of Squares 16 Find. List of Squares 16
17 Find. List of Squares 18 Find. List of Squares 17
19 Find. List of Squares 20 Find. List of Squares 18
21 Find. List of Squares 22 Find. List of Squares 19
23 Find. List of Squares Estimating Square Roots Return to Table of Contents 20
All of the examples so far have been from perfect squares. What does it mean to be a perfect square? The square of an integer is a perfect square. A perfect square has a whole number square root. You know how to find the square root of a perfect square. What happens if the number is not a perfect square? Does it have a square root? What would the square root look like? 21
Square Perfect Root Square 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225 Think about the square root of 50. Where would it be on this chart? What can you say about the square root of 50? 50 is between the perfect squares 49 and 64 but closer to 49. So the square root of 50 is between 7 and 8 but closer to 7. Square Perfect Root Square 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225 When estimating square roots of numbers, you need to determine: Between which two perfect squares it lies (and therefore which 2 square roots). Which perfect square it is closer to (and therefore which square root). Example: 110 Lies between 100 & 121, closer to 100. So 110 is between 10 & 11, closer to 10. 22
Square Perfect Root Square 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225 Estimate the following: 30 200 215 Another way to think about it is to use a number line. 4 8 9 2 3 Since 8 is closer to 9 than to 4, 8 is closer to 3 than to 2, so 8 2.8 23
24 The square root of 40 falls between which two perfect squares? 25 Which whole number is 40 closest to? 24
26 The square root of 110 falls between which two perfect squares? 27 Estimate to the nearest whole number. 110 25
28 Estimate to the nearest whole number. 213 29 Estimate to the nearest whole number. 90 26
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