## Miscellaneous Examples of Algebra: Part 3 for IITJEE Mains

Many identities can be readily established by making use of the properties of the cube roots of unity; as usual these will be denoted by .

**Problem 1:**

Show that

**Solution 1:**

The expression, E, on the left vanishes when ; hence, it must contain as a factor.

Putting , we have

Hence, E contains as a factor; and similarly, we may show that it contains as a factor; that is, E is divisible by

, or .

Further, E being of seven, and of five dimensions, the remaining factor must be of the form , thus,

.

Putting , we have ; putting , we have , and hence, ;

.

**Problem 2:**

Show that the product of and can be put in the form .

**Solution 2:**

The product

By taking these six factors in the pairs ,

,

and ,

we obtain the partial products:

, , and

where , , and

Thus, the product

which, in turn, equals

More esoteric algebraic miscellany is planned for you!

Nalin Pithwa

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