CBSE Class 8 Mathematics Cubes and Cube Roots Notes Set 04

Cubes and Perfect Cubes

The cube of a number is the number multiplied by itself three times, \( a^3 = a \times a \times a \). A natural number is a perfect cube if it is the cube of a natural number. A solid cube of side 1 cm repeated 3 x 3 x 3 times makes a big cube of side 3 cm, which is where the name comes from. There are ten perfect cubes from 1 to 1000 and four from 1 to 100 (1, 8, 27, 64).

Properties of Cubes

1. The cube of an even number is even and the cube of an odd number is odd.
2. The cube of a negative number is negative, because an odd number of negative factors gives a negative product: \( (-5)^3 = -125 \) and \( (-0.1)^3 = -0.001 \).
3. The unit digit of a cube depends only on the unit digit of the number: numbers ending in 2, 3, 7, 8 have cubes ending in 8, 7, 3, 2 respectively; endings 0, 1, 4, 5, 6, 9 give cubes ending in 0, 1, 4, 5, 6, 9.
4. In the prime factorisation of a perfect cube every prime occurs a multiple of 3 times. For example \( 64 = 2^6 \) is a cube, while \( 128 = 2^7 \) is not.

Patterns Linked to Cubes

Cubes are sums of consecutive odd numbers: \( 1^3 = 1 \), \( 2^3 = 3 + 5 \), \( 3^3 = 7 + 9 + 11 \), \( 4^3 = 13 + 15 + 17 + 19 \), \( 5^3 = 21 + 23 + 25 + 27 + 29 \). The cube of n is the sum of n consecutive odd numbers, so \( 10^3 \) needs ten odd numbers, 91 to 109.

Cube Root

The cube root undoes cubing: if \( 3^3 = 27 \) then \( \sqrt[3]{27} = 3 \), also written \( (27)^{1/3} \). For square roots the index 2 is usually omitted, but for cube roots the index 3 must always be written. Cube roots of negative numbers are negative, for example \( \sqrt[3]{-125} = -5 \).

Cube Root by Prime Factorisation

1. Break the number into prime factors by repeated division.
2. Group the equal factors in threes.
3. Take one factor from each group and multiply them.

Example: \( 1728 = 2^3 \times 2^3 \times 3^3 \) gives \( \sqrt[3]{1728} = 2 \times 2 \times 3 = 12 \). Example: \( 216 = 2^3 \times 3^3 \) gives 6.

Cube Root of a Perfect Cube by Estimation

1. Look at the unit digit of the cube to get the unit digit of the root (a cube ending in 6 has a root ending in 6; ending in 2 gives 8, and so on).
2. Strike off the last three digits and look at the number left. Find the two consecutive cubes between which it lies; the smaller base is the tens digit.

Example: 46656 ends in 6, so the root ends in 6. The remaining 46 lies between \( 3^3 = 27 \) and \( 4^3 = 64 \), so the tens digit is 3 and the root is 36. Example: 195112 ends in 2, so the root ends in 8; 195 lies between \( 5^3 = 125 \) and \( 6^3 = 216 \), so the root is 58. This method works only when the number is known to be a perfect cube.

Number of Digits in a Cube Root

Put a dot over the unit digit and then over every third digit moving left. The number of dots equals the number of digits in the cube root. So 1728 has two dots (two-digit root, 12), 8 has one dot (root 2), and 68921000 has three dots (root 410 has three digits). A perfect cube with 1, 2 or 3 digits has a one-digit root, with 4, 5 or 6 digits a two-digit root, and with 7, 8 or 9 digits a three-digit root.

Making a Number a Perfect Cube

1. To find the least multiplier, factorise the number and multiply by the primes needed to lift each exponent to the next multiple of 3. Example: \( 45 = 3^2 \times 5 \), so multiply by \( 3 \times 5^2 = 75 \) to get \( 3^3 \times 5^3 = 3375 \).
2. To find the least divisor, divide by the primes whose exponents are not multiples of 3. Example: \( 4394 = 2 \times 13^3 \), so divide by 2 to get \( 13^3 = 2197 \), whose cube root is 13.

Sums of Natural Numbers, Squares and Cubes

To add a block such as 10 + 11 + ... + 20, subtract: the sum of 1 to 20 is 210, the sum of 1 to 9 is 45, so the block is 165.

Worked Examples

1. Is 128 a perfect cube? \( 128 = 2^7 \) and 7 is not a multiple of 3, so it is not.
2. Find the cube of 0.9. \( 0.9 \times 0.9 \times 0.9 = 0.729 \).
3. Find \( \sqrt[3]{2744} \). Ends in 4 so the root ends in 4; 2 lies between \( 1^3 \) and \( 2^3 \), so the tens digit is 1; the root is 14.