Tricky Exponents!!

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!

2³ = 2 x 2  x 2

But this could also be written as 2³ = 2² 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, 2³ = 2² + 2².

In order to generalise it, 3⁵ = 3⁴ x 3, or 3⁴ + 3⁴ + 3⁴ or

5⁷= 5⁶ x 5 = 5⁶ + 5⁶ +5⁶ +5⁶ +5⁶

This property is extensively used in standardised tests:

1. If, 2^m + 2ⁿ  = 2⁴⁰ find the value of (m + n) , where, m & n are integers.
2. 40
3. 60
4. 78
5. 80
6. 158

Solution:

Using the same property, we have 2^m + 2ⁿ = 2⁴⁰ = 2³⁹ x 2³⁹.

This is the only combination whose sum will give 2⁴⁰.

Thus, m & n = 39

m + n  = 39 + 39 = 78

Answer: C

2. If, 3^x x 3^y x 3^z = 3¹², where x, y & z are integers. Find the highest prime factor of xyz.
3. 3
4. 5
5. 11
6. 13
7. 127

Solution:

Same logic, so, 3¹² = 3¹¹ + 3¹¹ + 3¹¹ ;

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

xyz = 11 x 11 x 11 = 11³

The highest prime factor of  is 11³ is 11.

Answer: C