In the article Simple Formula and their Applications I we dealt with algebraic formulas in the second degree, i.e., formulas related to perfect squares and the sum and difference of two squares. In this article we will be covering the algebraic formulas in the third degree, i.e., formulas related to perfect cubes and the sum and difference of two cubes.

**Formula:**

Again, [Taking common from ]

**Proof:**

**Corollary:**

or,

**Example: Find the cube of **

**Solution:**

**Example: Find the cube of **

**Solution:**

**Example: If , show that **

**Solution:**

[ ] (Proved)

**Example: Simplify **

**Solution:**

**Putting for and for we have,**

**The given expression **

Example: Show that when

Solution:

from the given condition we have,

(Proved)

**Formula: **

Again, [Taking common from ]

**Proof:**

**Corollary:**

or,

**Example: Find the cube of **

**Solution:**

**Example: Find the cube of **

**Solution:**

**Example: Find the cube of 297**

Solution:

[ ]

**Example: If , find the value of **

**Solution:**

…(1)

[ ]

[Using (1)]

**Exercise 1:**

1. Find the cube of:

a)

b)

c)

d)

2. Simplify:

a)

b)

c)

d)

3. Find the value of when and

4. Find the value of when and

5. If , show that,

6. If show that

7. If show that

8. If find the value of

**Formula:**

**Proof:**

**Conversely, . Hence, any expression of the form can be resolved into factors.**

**Example: Multiply by **

**Solution:**

Putting for and for we have,

Hence,

**Example: Resolve into factors **

**Solution:**

**Example: Resolve into factors **

**Solution:**

**Formula: **

**Proof:**

**Conversely, . Hence, any expression of the form can be resolved into factors.**

**Example: Multiply by **

**Solution:**

Putting for and for we have,

**Example: Resolve into factors **

**Solution:**

**Example: Resolve into factors **

**Solution:**

**Exercise 2:**

1. Multiply:

a) by

b) by

c) by

d) by

2. Resolve into factors:

a)

b)

c)

d)

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