In this blog, we will learn about cube and cube roots with the help of examples.
What is a cube?
A cube is a 3 dimensional figure having length = breadth = height. But in arithmetic cube means. When a number is multiplied three times by itself. i.e., for number 4 we have 4 x 4 x 4 = 43 = 64.
Multiplying 5 three times we get 5 x 5 x 5 = 53 = 125
Similarly, a x a x a = a3
is called "a cubed" or "a raised to the power of 3."
So to get a perfect cube we need to multiply a number by itself 3 times.
Properties Of Cube Numbers
Let's learn about the properties of cube numbers in this blog on cube and cube roots.
1. Cubes of all odd natural numbers are odd.
2. Cubes of all even natural numbers are even.
3. Cubes of negative integers are negative and cubes of positive integers are positive.
4. The cubes of numbers with unit digits 0, 1, 4, 5, 6, 9 will have the same unit digit.
The cube of numbers with unit digits:
2 will have unit digit 8.
8 will have unit digit 2.
3 will have unit digit 7. and
7 will have unit digit 3.
5. 1 and 8 are only cubes in one digit.
6. 27 & 64 are only cubes in two digits.
7. There are only 4 cubes between 1 to 100.
Cube and Cube Roots Examples
To understand cube and cube roots, let's dive into some examples.
1. Is 243 a perfect cube?
Solution: 243 = 3 x 3 x 3 x 3 x 3
In the above factorization, 3 × 3 remains after grouping the 3s in triplets. Therefore, 243 is
not a perfect cube.
2. Is 392 a perfect cube? If not, find the smallest natural number by which 392 must be multiplied so that the product is a perfect cube.
Solution: 392=2×2×2×7×7
The prime factor 7 does not appear in a group of three. Therefore, 392 is not a perfect cube. To make its a cube, we need one more 7.
In that case, 392×7=2×2×2×7×7×7=2744 which is a perfect cube. Hence the smallest natural number by which 392 should be multiplied to make a perfect cube is 7.
3. Is 53240 a perfect cube? If not, then by which smallest natural number should 53240 be divided so that the quotient is a perfect cube?
Solution: 53,240=23×5×113
The prime factor 5 does not appear in a group of three. So, 53240 is not a perfect cube.
In the factorization 5 appears only one time. If we divide the number by 5, then the prime factorization of the quotient will not contain 5.
So, 53240−5=2×2×2×11×11×11
Hence the smallest number by which 53240 should be divided to make it a perfect
cube is 5. The perfect cube, in that case, is = 10648.
4. Is 1188 a perfect cube? If not, by which smallest natural number should 1188 be divided so that the quotient is a perfect cube?
Solution: 1188=2×2×3×3×3×11
The primes 2 and 11 do not appear in groups of three. So, 1188 is not a perfect cube. In the factorization of 1188, the prime 2 appears only two times and the prime 11 appears once. So, if we divide 1188 by 2×2×112 \times 2 \times 112×2×11 equals 44, then the prime factorization of the quotient will not contain 2 and 11.
Hence the smallest natural number by which 1188 should be divided to make it a perfect cube is 44. And the resulting perfect cube is 1188 ÷ 44 = 27 (=33).
5. Is 68600 a perfect cube? If not, find the smallest number by which 68600 must be multiplied to get a perfect cube.
Solution: We have, 68600 = 2 x 2 x 2 x 5 x 5 x 7 x 7 x 7.
In this factorization, we find that there is no triplet of 5. So, 68600 is not a perfect cube.
To make it a perfect cube we multiply it by 5.
Thus,
2×2×2×5×5×5×7×7×7= 343000, which is a perfect cube
Read More: How To Find Cube Root: Prime Factorization and Estimation - Class 9
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