Square Numbers and Square Roots PDF

FOR ONLINE USE ONLY DO NOT DUPLICATE Chapter Ten Square numbers and square roots Introduction In this chapter, you will learn how to calculate square numbers not exceeding...

FOR ONLINE USE ONLY DO NOT DUPLICATE Chapter Ten Square numbers and square roots Introduction In this chapter, you will learn how to calculate square numbers not exceeding 10 000. You will also learn how to find the square root of square numbers LY of up to three digits, and to solve word problems involving square numbers and square roots. This competence will help you when dealing with large or small numbers, taking measurements, when describing computer’s N memory, dealing with growth of different things such as bacteria, finance, O interests and so on. Square numbers SE A square number is a number which is obtained by multiplying a number by itself. That is, a square of a number is the product of the given number and itself. U Finding square numbers by counting dots E Example 1 N LI Study carefully the arrangement of dots in the following table: (a) Write the number of dots in each set. N (b) Multiply the number of horizontal and vertical dots. (c) Write the product obtained after multiplying the horizontal and vertical O dots. R Multiply the number of Arrangement of dots Product horizontal and vertical dots FO 1×1 1 2×2 4 162 MATH std 5.indd 162 30/07/2021 14:49  FOR ONLINE USE ONLY DO NOT DUPLICATE 3×3 9 5×5 25 LY In this example, the square numbers are 1, 4, 9 and 25. N Finding square numbers using geometrical squares O Example 2 SE Study carefully the number of small squares in each of the following large squares, and then, find a relationship between the number of small squares U and square numbers. E 1×1=1 N 2×2=4 LI 3×3=9 N O R FO 4 × 4 = 16 5 × 5 = 25 163 MATH std 5.indd 163 30/07/2021 14:49  FOR ONLINE USE ONLY DO NOT DUPLICATE The number of small squares in each large square is a square number. Therefore, the square numbers are 1, 4, 9, 16, and 25. A number being multiplied by itself is said to be squared. For example, 5 ​× 5 = ​52 is five squared, 3​× ​3 = 32 is three squared, 4​× ​4 = 42 is four squared and so on. Example 3 LY Write the square number of each of the following numbers: (a) 7 (b) 10 (c) 11 N Solution O (a) ​7  ×  7  = 49​. Therefore, 49 is the square number of 7. (b) ​10  ×  10 = 100​. Therefore, 100 is the square number of 10. (c) ​11  ×  11  =  121​. Therefore, 121 is the square number of 11. Exercise 1 SE U Answer the following questions: 1. Find the product of each of the following numbers to obtain square numbers: E (a) ​20 × 20​ (b) ​21 × 21​ (c) 8 ​ × 8​ N (d) ​14 × 14​ (e) ​13 × 13​ LI 2. Write the square number of each of the following numbers: (a) 6 (b) 15 (c) 73 N (d) 9 (e) 31 O 3. Fill in the blanks in each of the following: (a) ​81 = 9 × ______ ​ R (b) ​144 = _____ × 12 ​ (c) 6 ​ 25 = 25 × ______ ​ FO (d) ​_____​  = 16 × 16 (e) ​961 = 31 × ______ ​ 164 MATH std 5.indd 164 30/07/2021 14:49  FOR ONLINE USE ONLY DO NOT DUPLICATE 4. Write the two missing square numbers in each of the following number patterns: (a) 1, 4, 9, 16, 25, _____, _____. (b) 1, 9, 25, 49, _____, _____. 5. Multiply each of the following numbers by itself and interpret the obtained product: (a) 53, 55, 57, 58 (b) 30, 33, 35 (c) 88, 89, 91 LY 6. List all the square numbers between 70 and 150. N 7. Write TRUE or FALSE against each of the following statements: O (a) All square numbers are divisible by 6. _____ (b) The product of any counting number by itself is a square number. _____ SE A number raised to two U A square number can be written as a number raised to two as follows: (a) ​9 = 3 × 3 = 3​2​​  ​​ is read as three squared or three raised to two. E (b) 4​ = 2 × 2 = 22 ​ is read as two squared or two raised to two. N (c) ​16 = 4 × 4 = 42 ​ is read as four squared or four raised to two. (d) ​49 = 7 × 7 = 72​ is read as seven squared or seven raised to two. LI Therefore, a number raised to two shows that it has been multiplied by itself. N O Example Write each of the following numbers as a number raised to two: R (a) 144 (b) 81 FO (c) 121 165 MATH std 5.indd 165 30/07/2021 14:49  FOR ONLINE USE ONLY DO NOT DUPLICATE Solution (​ a)    144 = 12 × 12 (b)     81 = 9 × 9                    =  ​12​​ 2​​ ​=  ​9​​ 2​​ ​(c)     121 = 11 × 11                   =  ​11​​ 2​​ LY Exercise 2 N Answer the following questions: 1. Write the following products as a number raised to two: O (a) ​6 × 6​ (b) ​9 × 9​ (c) 1​ 5 × 15​ (b) 256 (c) 121 SE 2. Write each of the following numbers as a number raised to two: (a) 225 (d) 289 3. Use the multiplication sign to represent each of the following numbers. U The first item is given as an example. (a) ​14 2 = 14 × 14​ (b) ​1 ​0​​  2​​= (c) ​3 ​7​​  2​​= E (d) ​1 ​8​​  2​​ = (e) ​2 ​6​​  2​​= N 4. Write each of the following numbers without finding the product: (a) Twenty-three raised to two. LI (b) Thirty-five raised to two. (c) Seventeen raised to two. N (d) Forty-eight raised to two. O 5. Fill the blanks with the missing numbers: (a) 2 × 2 = _____ = 4. R (b) _____ × _____ = _____ = 49. FO (c) _____ × _____ = 52 = _____. (d) _____ × _____ = 162 = _____. 166 MATH std 5.indd 166 30/07/2021 14:49  FOR ONLINE USE ONLY DO NOT DUPLICATE 6. Complete the following Table: Number 6 9 1 Number raised to 62 112 two Square 36 169 number LY 7. Find the products of the following: N (a) 192 (b) 362 (c) 812 O Square root of a number The square root of a given number is a number _ which when multiplied by SE itself gives a square number. The symbol ​​√​​  ​​ ​​ is called the square root and is used for finding the square root of a square number. For example; √ means square root of 49. _ ​​ 49 ​​ _ _ U ​​√ 49 ​​= √ ​​ 7 × 7 ​  = 7​. _ Therefore,​​√49 ​  = 7​. That is, the square root of 49 is 7. E N Example 1 LI Find the square root of each of the following numbers: (a) 25 (b) 9 (c) 100 N O Solution _ _ (a) ​​√25 ​​= √ ​​ 5 × 5 ​​= 5. Therefore, 5 is the square root of 25 or 52. R _ _ FO (b) ​​√ 9 ​​= √ ​​ 3 × 3 ​​= 3. Therefore, 3 is the square root of 9 or 32. 167 MATH std 5.indd 167 30/07/2021 14:49  FOR ONLINE USE ONLY _ _ DO NOT DUPLICATE (c) √ ​​ 100 ​​= ​​√10 × 10 ​​= 10. Therefore, 10 is the square root of 100 or 102. Example 2 Use a tree diagram to find the square root of 64. Solution 64 LY 2 32 N 2 16 O 2 8 2 SE 4 2 2 U Steps: _ 1. Write the square root of 64: That is,​​√64 ​​ _ ________________ 2. The product of the prime factors of 64: ​​√64 ​  = ​√2    × 2 × 2 × 2 × 2 × 2 ​​ E 3. Arrange _ the like factors in groups of two: ​​ ____________________ N √64 ​  = ​√_   ​    (2 × 2)​ × ​(2 × 2)​ × ​(2 × 2)​ ​​ _ _ = ​  ​√_ 2 × 2 ​​ × ​​√2 × 2 ​​ × ​​√2 × 2 ​​ LI 4. Since_ ​​√2 _ × 2 ​  = 2​,_ it follows _ that, ​​√64 ​  = ​√ 2 × 2 ​ × ​√2 × 2 ​ × ​√2 × 2 ​ N             =         2     ×      2        ×        2 O              =    8​. _ Therefore, ​​√ 64 ​  = 8​. R Example 3 FO _ Find ​​√676 ​​ 168 MATH std 5.indd 168 30/07/2021 14:49  FOR ONLINE USE ONLY DO NOT DUPLICATE Solution Steps: 1. Find the prime factors of 676 by division: 2 676 2 338 13 169 13 13 1 2. Express 676 as a product of its prime factors, that is, 6​ 76____________ = 2 × 2 × 13 × 13​ LY _ 3. The product of the prime factors of 676 is: _ ​​√676 ​  = ​_______________ √2   × 2 × 13 × 13 ​​ 4. Arrange the factors into two equal groups: √​​ 676 ​  = ​√(​   2 × 2)​ × ​(13 × 13)​ ​​ _ _ N ​ = ​(​√ 2 × 2 ​)​ × ​(​√13 × 13 ​)​​ O _ _ 5. Since ​​√2 × 2 ​  = 2​and √ _______________ ​​ 13 × 13 ​  = 13,​it follows that; ​​ √(​   2 × 2)​ × ​(13 × 13)​ ​ = 2 × 13 = 26​ _ Therefore, ​​√676 ​  = 26​. SE U Exercise 3 Answer the following questions: E 1. Find the square root of each of the following numbers: N (a) 196 (b) 400 (c) 121 (d) 324 LI 2. Find the square root of each of the following numbers: N (a) ​51 × 51​ (b) ​15 × 15​ O (c) ​73 × 73​ (d) ​99 × 99​ 3. Find the square root of each of the following numbers by using prime R factors: (a) 729 (b) 529 FO (c) 625 (d) 784 169 MATH std 5.indd 169 30/07/2021 14:49  FOR ONLINE USE ONLY DO NOT DUPLICATE 4. Use a tree diagram to find the square root of each of the following numbers: (a) 81 (b) 676 (c) 961 5. Find each _ of the following: _ (a) √ ​​ 256 ​​ (b) ​​√361 ​​ Word problems involving square number and square root of numbers LY Example N A lorry can carry 576 crates of soda. The crates are arranged in the lorry in such a way that there are equal rows and equal columns. How many O crates of soda will each row or each column have? Solution order _ SE Each row and each column has the same number of crates of soda. In to get the number of crates in each row and column, you have to find ​​√576 ​​. That is, U _ ________________________ ​​ 576 ​  = ​√2  ×  2  ×  2  ×  2  ×  2  ×  2  ×  3  ×  3 ​​ √     _______________________ =​  ​√(2    × 2)  (2 × 2)  (2 × 2)  (3 × 3) ​​ E =​ 2  ×  2  ×  2  ×  3​ N =​ 24​. LI Therefore, each row and each column will have 24 crates of soda. N Exercise 4 O Answer the following questions: 1. Find the number which is 13 raised to two. R 2. Kambi, Maria and Shelembi were required to write the number 25 in FO different _ ways. Kambi wrote 52, Maria wrote 5 × 5 and Shelembi wrote ​​ √625 ​​. Were their answers correct? Give reasons. 170 MATH std 5.indd 170 30/07/2021 14:49  FOR ONLINE USE ONLY DO NOT DUPLICATE 3. John’s age is the same as the square root of 100. What is the age of John in years? 4. Teacher Christina has 17 raised to 2 chickens. How many chickens does she have? 5. The age of a certain child multiplied by itself is the same as the age of the father. If the age of the father is 49 years, find the age of the child? LY Exercise 5 N Answer the following questions: 1. Fill in the blanks. O (a) 25, 36, 49, 64, _____, _____. (b) 121, 169, 225, 289, 361, _____, _____. 3 SE 2. Copy the following table and fill numbers in the blank spaces Number 23 31 Number raised to two 3 2 18 2 23 2 312 U Square number 9 441 625 3. Find the square roots of the following numbers: E (a) 1 (b) 49 (c) 225 (d) 400 N 4. List all the square numbers between 2 and 98. LI 5. Fill in the blanks in each of the following: N (a) ​400 = 20 × _____​. (b) _​ ____ = 15 × 15​. O (c) 1​ 69 = _____ × 13.​ R 6. Fill in the blanks in each of the following: (a) 16, 25, 36, 49, 64, _____, _____. FO (b) 64, 100, 144, _____, _____. (c) 25, 49, 81, _____, _____. (d) 225, 289, 361, 441, 529, _____, _____. 171 MATH std 5.indd 171 30/07/2021 14:49  FOR ONLINE USE ONLY DO NOT DUPLICATE 7. Find the square root of 841. _ _ _ _ 8. (a) Find the value of each of the following: √ ​​ 625 ​​, √ ​​ 576 ​​, √ ​​ 961 ​​ and √ ​​ 729 ​​. (b) Arrange the answers in part (a) from the smallest to the largest. 9. Use a tree diagram to find the square root of each of the following numbers: (a) 196 (b) 225 (c) 441 LY (d) 841 (e) 900 10. Find the square root of each of the following numbers by the prime N factors method: (a) 81 (b) 169 (c) 289 O (d) 256 (e) 576 SE 11. Which number when squared the result is two hundred and eighty nine? 12. The age of Tumaini is the same as the square root of one hundred and U forty four. What is the age of Tumaini in years? E Summary N 1. The number obtained by multiplying a number by itself is called a square number. LI 2. The square root of a given number is a number which when multiplied by itself gives_ a square number. N 3. The symbol ​​√ ​​  ​​ ​​represent the “square root” of a number. O R FO 172 MATH std 5.indd 172 30/07/2021 14:49