Exponents rules.

Exponents rules are also called laws of indices or power rules.

We will first define what an exponent is and then proceed to the list of Exponents rules with some examples.

What is an exponent?

3⁷=3 * 3 * 3 * 3 * 3 * 3 * 3

In the above equation in the Left Hand Side, 7 is the exponent and 3 is the base.

Hence, exponent is simply the number of times the base is multiplied by itself.

The word Exponent is often used interchangeably with power and index.

Do we actually need them?

As you can see from the equation, writing in the form of exponents is a convenient way of expressing a number which is multiplied by itself several times.

Can an exponent be only an integer?

No, not necessarily.

It can be a fraction also (It can be a negative integer also).

For example, when we say square root of 2, then it is equivalent to 2^1/2.

i.e. √2 = 2^1/2

Below, we will enumerate a few of the most frequently used Exponents Rules

Exponents rule number 1

x^m * xⁿ = x^m+n

In the product on the L.H.S., since the base is same i.e., x, so, the powers can be added.

For example, 

2⁵ * 2² = 2*2*2*2*2*2*2 = 2⁵⁺² = 2⁷

Exponents rule number 2

x^m/ xⁿ = x^m-n

when two exponents numbers with same base are divided by each other, then the result is the base raised to a power which is the difference of the two powers in the division.

For example,  

2⁵/2²=2*2*2*2*2/2*2= 2^5-2 = 2³

Exponents rule number 3  - 0 –

x^-n = 1/xⁿ

For example,

2^-3=1/2³

Thus, it follows that (x/y)^-n=(y/x)ⁿ

Exponents rule number 4

(x^m)ⁿ = x^m*n

For example,

(2³)² = (2*2*2)² = 2*2*2*2*2*2= 2^3*2 = 2⁶

A few standard results following from the above exponents rules are:

• a¹ = a
• a⁰ = 1
• 1ⁿ = 1
• 0ⁿ = 0, for n > 0

 (Why n > 0? Because, anything divided by zero is infinity!)

Infinity is not a number, but a mathematical concept. Infinity is the idea of a number greater than the greatest number possibly imaginable).

We will solve some examples using the above exponents rules

Problem 1

Solve for n in the following exponential sum:

2ⁿ + 2ⁿ + 2ⁿ + 2ⁿ = 8³

Solution:

It is necessary that the base has to be equal to compare exponents on the two sides of the equation.

So, we first simplify the two sides of the equation using the exponents rules on each side.

LHS:  2ⁿ+2ⁿ+2ⁿ+2ⁿ = 4*2ⁿ = 2² *2ⁿ = 2^{n + 2}

RHS: 8³=(2³)³=2⁹

Now, since the bases are same on the both the sides,we can now equate the exponents on the two sides.

So, we have

n +2 = 9

Therefore,

 n=7.
Problem 2

If 3^k=7, What is the value of 3^{2k + 1} ?

Solution:

We simply apply the exponents rules directly as given below

3^2k+1=3^2k * 3 = (3^k)² * 3 =7² * 3 = 49 *3 =147

Problem 3

Solve (5⁶ – 5⁵)/ 25.

Solution:

We first simplify the numerator
5⁶- 5⁵ = 5 * 5⁵ - 5⁵ = 5⁵ (5 - 1) = 5⁵ * 4
Similarly in the denominator,
25 = 5²
Thus, we have
5⁵ * 4 / 5² = 5³ * 4 = 125*4 = 500

Problem 4

Find the value of 12(3^-1  –  4^-1)^-1

Solution:

We apply the exponents rules to the problem step by step to arrive at the following result
12(3^-1 - 4^-1)^-1 =
12(1/3 - 1/4)^-1 =
12(4/12 - 3/12)^-1   =
12(1/12)^-1 =
12*12 =144