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Squares and Square Roots Question 1) Write the squares of the numbers 1 to 20 in a table. Number 1 2 3 4 5 6 . . . Square The next point is an extremely important idea to remember for all areas of maths. Everything in maths has an opposite Although you have probably never talked about this directly you probably know this is true. Question 2) a) What is the opposite of multiplying? b) What is the opposite of addition? c) What is the opposite of subtraction? d) What is the opposite of division? It would therefore make sense that we much have an opposite to squaring, but what would it be? You’ve probably already heard of it before. Question 3) What do you think the opposite of squaring might be? The opposite of a square is a; Square Root Finding the square root of a number finds what number has to be squared to get the number you have. For example. The square root of 9 is 3, since 3 times 3 is 9. The square root of 25 is 5, since 5 times 5 is 25. The symbol we use to represent a square root is; √ And we write the number we have inside the square root sign. √9 Question 4) Write in a table all of the square roots of the numbers from 1-25, (you can use a calculator to help you) You’ll notice that there are some numbers in your table whose square roots are whole numbers. These numbers are called perfect squares. Question 5) What are the first 15 perfect squares? Using our knowledge of perfect squares we can estimate the values of the square roots of non-perfect squares. For example if we want to find √13 we need to look at which two perfect squares it falls between, in this case 9 and 16, with √9 = 3 πππ √16 = 4. Since 13 is closer to 16 than 9 the square root will be closer to 4 than 3. So a reasonable estimation would be that √13 = 3.6. Given the actual answer is 3.605, 3.6 is a pretty good estimate. Question 6) Estimate the following values a) √28 b) √45 c) √50 d) √60 e) √96 f) √140 g) √157 h) √200 Question 7) Check your estimates to question 16 using a calculator and write the answer correct answer to 2 decimals places. Question 8) Let’s try and get your estimates even closer. Estimate the square roots of the following numbers to two decimal places. Write the answer correct to 3 decimal next to your estimate. a) 7 b) 17 c) 21 d) 33 e) 42 One of the main purposes of using square roots to be able to calculate the side length of a square if you know the area. Question 9) a) If there area of a square is 16cm2, what is the length of the sides of the square? b) If there area of a square is 64cm2, what is the length of the sides of the square? c) If there area of a square is 115cm2, what is the length of the sides of the square? *Extension* Question 1) If a number multiplied by itself is called squaring, what is it called when you multiply a number by itself and by itself again, so 3 x 3 x 3? It is in fact called cubing or finding the cube of a number. So if you cube 3 then you write it as 33 to signify 3 x 3 x 3. Question 2) So squaring is related to finding the area of a square, so what do you think cubing might be related to finding? Question 3) Write the cubes of the numbers 1 to 10 in a table. Number Cube 1 2 3 . . 10 As we discussed earlier everything in mathematics has an opposite, and just like the opposite of squaring is square rooting, the opposite of cubing is cube rooting. The symbol we use to represent cube rooting is very similar to the square root sign expect it has a little 3 just above the “tick” of the square root sign as shown below. 3 √ So if you had trouble with question 2, I’ll put you out of your misery, finding the cube is related to finding the volume of a cube. Remember to find the volume of a cube or rectangular prism we multiply the length by the width by the height (we’ll cover this in more detail later) But remember a cube is special because the length, width and heights are all the same. Just as with square roots, cube roots also have a role to play, they are used when we know the volume of a cube and want to find the length of the sides. Question 4) Find the cube roots of the numbers from 1 to 30 and write them in a table. Number Cube root 1 2 . . . 30 Just like for square roots you’ll notice that some of the values for cube roots are whole numbers. These are called perfect cubes. In question 2 you found the first 10 perfect cubes. Use these results to help answer the questions below. Question 5) Estimate the values of the following cube roots. a) 10 b) 30 c) 60 d) 100 e) 500 f) 630 g) 800 Question 6) Check your estimates to question 5 by writing the exact value of the cube roots next to your estimates.