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In some previous articles we have learnt about a few interesting concepts on exponents. In this piece of articulation let us enjoy some exam focused properties of the tricky exponent!

2^{3} = 2 x 2 x 2

But this could also be written as 2^{3} = 2^{2} x 2; also, multiplication is addition that many number of times. Eg: 2 x 3 = 2 + 2 + 2.

Likewise, for the first equation, we can write, 2^{3} = 2^{2} + 2^{2}.

In order to generalise it, 3^{5} = 3^{4} x 3, or 3^{4} + 3^{4} + 3^{4} or

5^{7}= 5^{6} x 5 = 5^{6} + 5^{6} +5^{6} +5^{6} +5^{6}

This property is extensively used in standardised tests:

- If, 2
^{m}+ 2^{n}= 2^{40}find the value of (m + n) , where, m & n are integers. - 40
- 60
- 78
- 80
- 158

**Solution: **

Using the same property, we have 2^{m} + 2^{n} = 2^{40} = 2^{39} x 2^{39}.

This is the only combination whose sum will give 2^{40}.

Thus, m & n = 39

m + n = 39 + 39 = 78

**Answer: C**

- If, 3
^{x}x 3^{y}x 3^{z}= 3^{12}, where x, y & z are integers. Find the highest prime factor of xyz. - 3
- 5
- 11
- 13
- 127

**Solution: **

Same logic, so, 3^{12} = 3^{11} + 3^{11} + 3^{11} ;

Thus, possible values for x,y & z = 11, 11 and 11.

xyz = 11 x 11 x 11 = 11^{3}

The highest prime factor of is 11^{3} is 11.

**Answer: C**