Squares, square roots, cubes and cube roots

1 16 TOPIC 2 PAIRS quares, square roots , cubes and cube rootsBy the end of this topic, you should be able to: Find squares, square roots , cubes and cube roots of positive whole numbers, decimals and common fractionsUse a calculator to find squares, square roots , cubes and cube roots of any squares, square roots , cubes and cube rootsDiscuss what you remember about squares, square roots , 1 cubes and cube an example of each and compare your examples with 2 those of your activityYou can use a calculator to fi nd squares, square roots , cubes and cube roots . __ 100 x2 __ 2ndFyx4 x2yx4 32ndF __ 125 17 TOPIC 2 Squares, square roots , cubes and cube rootsSO ; ; numbers and cube numbersTo find a square number, multiply a number by itself. For example, 16 is a square number because 4 4 = 16.

2 To find a cube number, multiply a number by itself twice. For example, 27 is a cube number because 3 3 3 = conceptKey conceptLearn that square numbers and cube numbers are special cases of whole the square roots of square numbers and cube numbersGive examples of squares, square roots , cubes and cube Draw a number square from 1 to 100 in your exercise Draw a box around each square number in your 3 number all the square numbers as a number ) Count the number of members between each pair of b) square numbers. Check that these numbers form a number the rule of this number sequence to find the next c) two square numbers in the sequence. Find a way to check your a number square from 1 to 100 in your exercise 4 book. Draw a circle around each cube number in your number ) Write all the cube numbers as a number ) Write the next three numbers in this 1 square numbersSquares and square roots of whole numbersWe also use index notation to write square numbers.

3 For example, 9 9 = 92 and we read 92 as 9 squared .If you multiply a given number by itself to get a square number, that given number is a square root . For example, the square roots of 16 are 4 and 4 because 4 4 = 16 and 4 4 = 16. We use the symbol __ for square root , so ___ 16 = 4 and ___ 16 = 4. If there is no or sign, assume a sign, for example, ___ 25 = mental maths, pen-and-paper methods or a calculator to find the square of each 5 2 9 3 13 4 26 5 37 Solutions1 25 2 81 3 169 4 676 5 1 369 Example 118 TOPIC 2 Squares, square roots , cubes and cube rootsSO ; and square roots of common fractionsTo find the square of a common fraction, multiply the common fraction by itself. For example 1 __ 2 1 __ 2 or ( 1 __ 2 ) 2 = 1 __ 4 . To multiply two fractions, multiply the numerators with each other and the denominators with each find the square root of a common fraction, find the square root of the numerator and the a calculator to find the square of key sequence for a simple calculator is214 214 The key sequence for a calculator with an x2 function is214x2 2142 = 45 7961 Find the square of 2 __ 3.

4 2 Find the square root of 9 __ 64 .Solutions1 2 __ 3 2 __ 3 = 4 __ 9 2 ___ 9 __ 64 = 3 __ 8 Example 2 Example 3 Find squares and square roots of numbersFind the square of each a) 1 b) 2 c) 3 d) 4 e) 6 Use mental maths or pen-and-paper methods to find the 2 square of each ) 7 b) 8 c) 12 d) 10 e) 11 Use a calculator to find the square of each a) 29 b) 52 c) 88 d) 277 e) 3 914 Find the square root of each a) 1 b) 4 c) 9 d) 16 e) 25 Use mental maths to find the square root of each a) 64 b) 100 c) 49 d) 81 e) 121 f) 25 Find the missing number in each a) __ 4 = .. b) ___ 25 = .. c) ___ 64 = ..d) ____ 100 = .. e) ____ 144 = .. f) ___ 49 = ..Exercise 119 TOPIC 2 Squares, square roots , cubes and cube rootsSO ; and square roots of decimalsTo find the square root of a common fraction that is not a perfect square , convert it to a decimal fraction.

5 Then use a calculator with a square root function, for example __ 2 __ 3 _____ the answer so that you can check the calculation. For example, __ 5 should be between 2 and 3 because 22 = 4 and 32 = 9. So __ 5 = makes the square of each decimal correct to two decimal places, where 2 3 4 5 2 3 1 5 30 the square root of each decimal correct to two decimal places. Use a calculator to find the square root , then use your estimates to check the _____ 2 ____ 3 _____ 4 _____ 5 _____ Solutions1 2 3 4 5 4 Example 5 Find squares and square roots of common fractionsFind the square of each a) 2 __ 5 b) 3 __ 4 c) 3 __ 5 d) 2 __ 7 e) 5 __ 8 Find the missing number in each a) ( 3 __ 8 ) 2 =.

6 B) ( 1 __ 2 ) 2 = .. c) ( 4 __ 9 ) 2 = ..d) ( 1 __ 5 ) 2 = .. e) ( 2 __ 7 ) 2 = .. f) ( 3 __ 5 ) 2 = ..Find the square root of each a) 4 __ 9 b) 1 __ 4 c) 9 __ 16 d) 4 __ 25 e) 25 __ 36 Find the missing number in each a) __ 1 __ 4 = .. b) __ 1 __ 9 = .. c) ___ 16 __ 49 = ..d) ___ 9 __ 64 = .. e) ___ 25 __ 81 = .. f) ___ 49 __ 25 = ..Exercise 2 NoteWhen a question does not state to how many decimal places you must round the answer, round the answer to two decimal 2 Squares, square roots , cubes and cube rootsSO ; ; squares and square roots of decimalsUse a calculator to find the square of each decimal correct 1 to two decimal ) b) c) d) e) a calculator to find the square of each decimal correct 2 to two decimal places, where ) b) c) d) e) the missing number in each case.

7 Give the answers 3 correct to two decimal places, where ) = .. b) = .. c) = ..d) = .. e) = .. f) = ..Estimate the square root in each case. Then use a calculator 4 to find the square root correct to two decimal ) b) c) d) e) the missing number in each case correct to two 5 decimal ) _____ = .. b) _____ = .. c) _____ = ..d) _____ = .. e) _____ = .. f) _____ = ..Exercise 3 Cube numbersCubes and cube roots of whole numbers and decimalsWe also use index notation to write cube numbers. For example, 2 2 2 = 23 and we read 23 as 2 to the power of 3 .If you multiply a given number by itself twice to get a cube number, that given number is a cube root . For example, the cube root of 27 is 3 because 3 3 3 = 27. We use the symbol 3 __ for cube root , so 3 ___ 64 = 4 and 3 ___ 43 = need a calculator with 2ndF and yx keys to find cubes and cube roots in difficult cases.

8 You can also write numbers as products of their prime factors to help you find cube conceptDetermine the cube roots of a calculator to find the cube of key sequence for a simple calculator 11 . 3 The key sequence for a calculator with a yx function = 1 621 TOPIC 2 Squares, square roots , cubes and cube rootsSO ; ; 216 as the product of its prime factors, then find its cube 216 Divide 216 by 2 and write 108 on the next 108 Divide 108 by 2 and write 54 on the next 54 Divide 54 by 2 and write 27 on the next 27 Divide 27 by 3 and write 9 on the next 9 Divide 9 by 3 and write 3 on the next 1216 = 2 2 2 3 3 3 = 23 33 3 ____ 216 = 3 _____ 23 33 = 2 3 = 6 Estimate the cube root of and explain your decision. Use a calculator to find the value correct to two decimal places.

9 Use your estimate to check the because 3 _____ should be between 3 and 4. 33 = 27 and 43 = 64, so the answer is closer to 3 than 4 because is closer to 27 than to key sequence is3 3 _____ = answer of makes sense because it is close to the 7 Example 8 Find cubes and cube roots of whole numbers and decimalsUse pen-and-paper methods to find the cube of each 1 ) 7 b) 9 c) 13 d) 18 e) 20 Use a calculator to find the cube of each decimal correct to 2 two decimal ) b) c) d) e) a calculator to find the missing number in each case 3 correct to two decimal ) = .. b) = .. c) = ..d) = .. e) = .. f) = ..Exercise 4 NoteEstimating cube roots helps us to check calculator issueIn engineering, we use square roots to calculate the radii of circles and the sides of squares.

10 We also use cube roots to calculate the radii of spheres and the sides of 2 Squares, square roots , cubes and cube rootsSO ; ; and cube roots of common fractionsTo find the cube of a common fraction, multiply the common fraction by itself twice. For example, 1 __ 2 1 __ 2 1 __ 2 or ( 1 __ 2 ) 3 = 1 __ 8 . To multiply fractions, multiply the numerators with one another and the denominators with one find the cube root of a common fraction, find the cube root of the numerator and the each number as the product of its prime factors. 4 Then find the cube root of each ) 27 b) 64 c) 125 d) 343 e) 729 Estimate the cube root of each decimal, then use a 5 calculator to find the value correct to two decimal places. Use your estimates to check the ) b) c) d) 120 e) the cube of 1 2 __ 5.