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NUMBER AND ALGEBRA • NUMBER AND PLACE VALUE 3C Squares and square roots Square numbers ■ Square numbers are numbers that can be arranged in a square, as shown below. 12 = 1 22 = 4 32 = 9 42 = 16 ■ ■ ■ ■ ■ The first square number, 1, equals 1 ì 1. The second square number, 4, equals 2 ì 2. The third square number, 9, equals 3 ì 3. The fourth square number, 16, equals 4 ì 4. If this pattern is continued, any square number can be found by multiplying the position of the square number by itself. This is known as squaring a number and is written by an index (or power) of 2. For example, 3 ì 3 can be written as 32. WORKED EXAMPLE 10 Find the square numbers between 90 and 150. THINK 1 Use your knowledge of tables to find the first square number after 90. 102 = 10 ì 10 = 100 2 Find the square numbers which come after that one but before 150. 112 = 11 ì 11 = 121 122 = 12 ì 12 = 144 132 = 13 ì 13 = 169 3 78 WRITE Write the answer in a sentence. Maths Quest 7 for the Australian Curriculum (too big) The square numbers between 90 and 150 are 100, 121 and 144. NUMBER AND ALGEBRA • NUMBER AND PLACE VALUE WORKED EXAMPLE 11 Write the two whole numbers between which 5.72 will lie. THINK WRITE 1 Write the whole numbers either side of 5.7. 5.7 is between 5 and 6. 2 Consider the square of each whole number. 5.72 is between 52 and 62. 3 Simplify 52 and 62 then write the answer in a sentence. So 5.72 is between 25 and 36. 4 Verify your answer with a calculator. Square roots ■ ■ ■ ■ Evaluating the square root of a number is the opposite of squaring a number. By writing 42 = 16, we mean that 4 multiplied by itself is equal to 16. The opposite of this is to say that the square root of 16 is 4, or 16 = 4. Using the illustrations of the squares above, a square of area 16 square units must have a side length of 4 units. To find the square roots of larger numbers, it helps to break the number up as a product of two smaller square roots with which we are more familiar. For example, 900 = 9 ì 100 = 3 ì 10 = 30 WORKED EXAMPLE 12 Find: a 49 b 3600 . THINK a b Find a number which when multiplied by itself gives 49 (7 ì 7 = 49). WRITE a b 49 = 7 1 Write 3600 as the product of two smaller numbers for which we can find the square root. 2 Take the square root of each of these numbers. = 6 ì 10 3 Find the product. = 60 3600 = 36 ì 100 WORKED EXAMPLE 13 Between which two numbers will 74 lie? THINK 1 Write the square numbers either side of 74. 2 Consider the square root of each number. 3 Simplify 64 and 81. 4 Verify your answer with a calculator. WRITE 74 is between 64 and 81. 74 is between 64 and 81. So 74 is between 8 and 9. Chapter 3 Indices and primes 79 NUMBER AND ALGEBRA • NUMBER AND PLACE VALUE REMEMBER 1. Square numbers are written using an index (or power) of 2 and are found by multiplying the number by itself. 2. The opposite of squaring a number is finding the square root of a number. 3. To find the square root of a number we can use our knowledge of square numbers or use a calculator. EXERCISE 3C INDIVIDUAL PATHWAYS eBoo k plus eBook Activity 3-C-1 Squares and square roots doc-1681 Activity 3-C-2 More squares and square roots doc-1682 Activity 3-C-3 Advanced squares and square roots doc-1683 Squares and square roots FLUENCY 1 Evaluate and verify your answers with a calculator. a 82 b 112 c 152 d 252 2 WE10 a Find the square numbers between 50 and 100. b Find the square numbers between 160 and 200. 3 WE11 Write two whole square numbers between which each of the following will lie. a 6.42 b 7.82 c 9.22 d 12.52 4 WE12a Find: a 25 81 b c 144 d 400 d 160 000 5 WE12b Find and verify your answers with a calculator. a 4900 b 14 400 c 360 000 6 WE13 Between which two numbers will 60 lie? 7 Write the two whole numbers between which each of the following will lie. a 14 b 90 c 200 2 d UNDERSTANDING 8 a Find the even square numbers between 10 and 70. b Find the odd square numbers between 50 and 120. 9 MC For which of the following square roots can we calculate an exact answer? A 10 B 25 C 50 D 75 E 82 10 MC For which of the following square roots can we NOT calculate the exact value? A 160 B 400 C 900 D 11 Evaluate the following. Verify your answers with a calculator. a 22 + 25 b 92 - 36 c 52 ì 22 ì 49 d 32 + 22 ì 16 e 32 - 22 ó f 80 4 + 49 9 ì 42 - 144 ó 22 Maths Quest 7 for the Australian Curriculum 2500 E 3600