# Cube of Numbers:

#### Overview:

 Topic Included: Formulas, Definitions & Exmaples. Main Topic: Quantitative Aptitude. Quantitative Aptitude Sub-topic: Simplification Aptitude Notes & Questions. Questions for practice: 10 Questions & Answers with Solutions.

Cube of numbers always help during the solving the questions of simplification, so try to remember the cube of numbers at least till 25. If you remember the cubes, it will help you to speed up your solving capacity. If you can't remember then use the method given below.

 $$1^3$$ 1 $$2^3$$ 8 $$3^3$$ 27 $$4^3$$ 64 $$5^3$$ 125 $$6^3$$ 216 $$7^3$$ 343 $$8^3$$ 512 $$9^3$$ 729 $$10^3$$ 1000 $$11^3$$ 1331 $$12^3$$ 1728 $$13^3$$ 2197
 $$14^3$$ 2744 $$15^3$$ 3375 $$16^3$$ 4096 $$17^3$$ 4913 $$18^3$$ 5832 $$19^3$$ 6859 $$20^3$$ 8000 $$21^3$$ 9261 $$22^3$$ 10648 $$23^3$$ 12167 $$24^3$$ 13824 $$25^3$$ 15625

Method to find the cube of numbers: The cube of numbers can find out by using the algebraic formulae given below.$$(a + b)^3 = a^3 + b^3 + 3ab \ (a + b)$$ $$(a - b)^3 = a^3 - b^3 - 3ab \ (a - b)$$ To find the cube of any number you have to use only one formula out of two which are given above. Let's take some examples to understand the method.

Example(1): Find the cube of 14?

Solution: The number 14 can be written as (10 + 4) or (20 - 6) and the cube of 14 can find out two ways.

If you are finding cube of 14 by using (10 + 4), then use the formula $$(a + b)^3 = a^3 + b^3 + 3ab \ (a + b)$$.

If you are finding cube of 14 by using (20 - 6), then use the formula $$(a - b)^3 = a^3 - b^3 - 3ab \ (a - b)$$.

Method(1):$$14 = (10 + 4)$$ $$(a + b)^3 = a^3 + b^3 + 3ab \ (a + b)$$ Here, take a = 10, and b = 4.$$(10 + 4)^3$$ $$= 10^3 + 4^3 + 3 \times 10 \times 4 \ (10 + 4)$$ $$= 1000 + 64 + 120 \times 14$$ $$= 1064 + 1680 = 2744$$

Method(2):$$14 = (20 - 6)$$ $$(a - b)^3 = a^3 - b^3 - 3ab \ (a - b)$$ Here, take a = 20, and b = 6.$$(20 - 6)^3$$ $$= 20^3 - 6^3 - 3 \times 20 \times 6 \ (20 - 6)$$ $$= 8000 - 216 - 360 \times 14$$ $$= 8000 - 216 - 5040$$ $$= 8000 - 5256 = 2744$$ You have to choose the only one method to find the cube of any number.

Example(2): Find the cube of 25?

Solution: The number 25 can be written as (20 + 5) or (30 - 5) and the cube of 25 can find out two ways.

If you are finding cube of 25 by using (20 + 5), then use the formula $$(a + b)^3 = a^3 + b^3 + 3ab \ (a + b)$$.

If you are finding cube of 25 by using (30 - 5), then use the formula $$(a - b)^3 = a^3 - b^3 - 3ab \ (a - b)$$.

Method(1):$$25 = (20 + 5)$$ $$(a + b)^3 = a^3 + b^3 + 3ab \ (a + b)$$ Here, take a = 20, and b = 5.$$(20 + 5)^3$$ $$= 20^3 + 5^3 + 3 \times 20 \times 5 \ (20 + 5)$$ $$= 8000 + 125 + 300 \times 25$$ $$= 8125 + 7500 = 15625$$

Method(2):$$25 = (30 - 5)$$ $$(a - b)^3 = a^3 - b^3 - 3ab \ (a - b)$$ Here, take a = 30, and b = 5.$$(30 - 5)^3$$ $$= 30^3 - 5^3 - 3 \times 30 \times 5 \ (30 - 5)$$ $$= 27000 - 125 - 450 \times 25$$ $$= 27000 - 125 - 11250$$ $$= 27000 - 11375 = 15625$$ You have to choose the only one method to find the cube of any number.