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Chapter -01, Cubes and Cubc Reols., Introduction, Perfect Cubes - The cube of a number is the number raised to the power 3, Volume of Cube = 5X5X5 = 5, when a number is multiplied by itself 3 times. We get the cube of the number, The cube of a 2= 23= 2x2x2=8, The cube of 5= 53= 5x5x5 = 125, , 8 and 125 are natural numbers which are cubes of natural number 2 and 5 respectively, Such numbers are called cubic numbers., , Example:-1) What is the least number that 500 must be multiplied with to make "a, perfect cube?, , => We have seen that 500 = (2X2) X (5X5X5), = (2x2)x(5X5X5), =(2x2x2) x (5X5X5), =(23)x(59), , Hence, to make the factor into a product of triples of equal factor, 500 should be, multiplied by 2., , Example -- 2) What is the smallest number that 686 must be divided by, to make it a, perfect cube., , =>, , Factorising 686, we get 686 = 2x 7x 7x7, , , , , , , , The factor 7 is a triple, but 2 is alone 2 \ G sc, If we divide by 2, the number obtained +\S42., will be a perfect cubs. > Fq, 686 must be divided by 2 to make it is a +, , perfect cube., Properties of cubes of Numbers, , Cube of Even and Odd Numbers:, co Edit with WPS Office

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+ If an even number is multiplied by itself three times, the result is an even number, , Examples:- 2x2x2 =8, 6X6X6 = 216, 12X12X12 = 1728, Also when an odd number is multiplied by itselfy 3 times... the result is an odd number., Examples:- 3x3x3 = 27, 7X7X7 =343, 11X11X11 =1331, e The cube of an even natural number is even., e The cube of an odd natural number is odd., Cube of a Negative Integer:Look at the products given below, (1)? = 1) X (1) X (1) = (1), (-5)? = (5) X (5) X (5) = (125), (-10)? =(-10) X (-10) X (-16) = (-1000), We conclude that, The cube of a negative integer is negative., Cube Roots:, If the cube of 2 is 8, then 2 is said to be the cube root of 8. We denote cube root by the, symbol °v, , Thus, *V8= 2, 3V27 = 3, 3V64= 4, #¥1000 = 10, , Cube root - The cube root of a given number is that number whose cube is equal to the, given number., , Cube Root of a Perfect Cube by Prime Factorisation., Example: Find the cube root of 1728., , => Carry out the prime factorisation of the given number., 1728 = 2X2X 2X 2X 2X 2X3 X3X3, , Make triples of equal factors., , =(2x2x2)x (2x2X2)X(3X3X3), , Take one prime factor from each triple and find their product, , 2X2X3 = 12, , co Edit with WPS Office