Cubes and Cube Roots
ICSE Class 8 · Mathematics · Chapter 4
What this chapter covers
- Cubes and perfect cubes
- Properties of cubes
- Patterns in cube numbers
- Cube roots
- Cube root by prime factorisation
- Making a number a perfect cube
- Cube roots of negative numbers and fractions
- Estimating cube roots
Key points
- The cube of a number is the product of three factors, each equal to that number: n3 = n × n × n.
- A perfect cube is a natural number that is the cube of a natural number, such as 1, 8, 27, 64 and 125.
- Cubes of even numbers are even and cubes of odd numbers are odd.
- The unit digit of a cube depends only on the unit digit of the number; 2 and 8 swap, 3 and 7 swap, and the other digits stay the same.
- A number ending in k zeros has a cube ending in 3k zeros.
- The cube of a negative number is negative, and the cube of a fraction is the cube of its numerator over the cube of its denominator.
- The sum of the right group of n consecutive odd numbers is n3, for example 7 + 9 + 11 = 27.
- The cube root of a number is the number whose cube gives that number, written sqrt[3]phantomx.
- To find a cube root by prime factorisation, group the prime factors in triplets and multiply one factor from each triplet.
- A factor left over outside a triplet means the number is not a perfect cube; multiply by the missing factors or divide out the extra ones to make it one.
- sqrt[3]-a = -sqrt[3]a, and the cube root of a fraction is the cube root of the numerator over the cube root of the denominator.
- To estimate a cube root, find the two perfect cubes the number lies between.
Questions answered
Find the cube of 9 and of 15.
93 = 729 and 153 = 15 × 15 × 15 = 225 × 15 = 3375.
Is the cube of 26 even or odd? What is its unit digit?
26 is even, so its cube is even. A number ending in 6 has a cube ending in 6.
Explain why 4900 cannot be a perfect cube.
4900 ends in exactly two zeros. A perfect cube ends in a multiple of three zeros, so 4900 is not a perfect cube.
Find (-(3)/(4))3.
(-(3)/(4))3 = ((-3) × (-3) × (-3))/(4 × 4 × 4) = (-27)/(64).
Use prime factorisation to find sqrt[3]125 and sqrt[3]343.
125 = 5 × 5 × 5, so sqrt[3]125 = 5. 343 = 7 × 7 × 7, so sqrt[3]343 = 7.
Is 250 a perfect cube? If not, find the smallest number by which it must be divided to make it one.
250 = 2 × 5 × 5 × 5. The 2 is left over, so 250 is not a perfect cube. Dividing by 2 gives 125 = 53.
Find the smallest number by which 256 must be divided to give a perfect cube, and find the cube root of the result.
256 = 28, which is two triplets of 2 with two 2s left over. Divide by 2 × 2 = 4 to get 64 = 43, so the cube root is 4.
Find sqrt[3]-512 and sqrt[3](8)/(125).
sqrt[3]512 = 8, so sqrt[3]-512 = -8. sqrt[3](8)/(125) = (2)/(5).