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CUbES AND CUbE rootS

Cubes Cube roots

• A number when multiplied by itself thrice. • The cube root of a number is the value which
• The cube of x = x × x × x = x 3 when multiplied by itself thrice produces the
• Cubes of positive integers are always positive. given number.
• Cubes of negative integers are always negative. • Finding the cube root is the inverse operation of
finding the cube of a number.

Perfect cubes • The cube root of any positive number is always
A natural number ‘n’ is called a perfect cube if there positive.
exists a natural number ‘m’ such that n = m × m × m • The cube root of any negative number is always
= m . negative.
3
• The symbol for cube root is 3 .
Properties of Perfect Cubes
• The cubes of all odd numbers are odd, and the Methods of Finding Cube root of a Number
cubes of all even numbers are even. • Prime Factorisation Method.
• The number of zeros at the end of a perfect cube • Estimation Method or using units digit and tens
is always a multiple of 3. digit.
• The cube of the numbers having digits 1, 4, 5, 6 • Successive Subtraction Method.
and 9 at their ones places are number ending in • Division Method.
the same digits respectively.
• The cube of a number ending with 2 ends with
digit 8 at its ones place and its vice versa. Properties of Cube roots
3
3
• The cube of a number ending with 3 ends with • If a = n , then a = n.
digit 7 at its ones place and its vice versa. • ab = 3 a × 3 b, where ‘a’ and ‘b’ are integers.
3
• The product of cubes of any two numbers is a 3 a
equal to the cubes of their product. • 3 = 3
• The sum of the cubes of first n natural numbers b b
3
3
is equal to the square of their sum. • −a = –a

Mental Maths

1. Write the cubes of the first five natural numbers.
2. Is 2000 a perfect cube?
3. The smallest number by which 4 should be multiplied to get a perfect cube is ............... .
4. The smallest number by which 81 should be divided to get a perfect cube is ............... .
5. The cube of 0.1 is ............... .
6. The cube root of 27 × 8 is ............... .
7. The cube root of 0.343 is ............... .
8. What will be the volume of a cube if the length of its edge is 11 cm?
9. The cube of an odd natural number is always ............... .
10. Observe the following pattern:
3
3
3
3
2
3
2 – 1 = 1 + 2 × 1 × 3; 3 – 2 = 1 + 3 × 2 × 3; 4 – 3 = 1 + 4 × 3 × 3
Using this pattern, find the value of each of the following.
3
3
3
(a) 75 – 74 (b) 111 – 110 3
161 Cubes and Cube Roots