An Introduction to Exponents

How about calculating the distance between moon and earth? You will be surprised that the distance is even greater than lakhs. The actual distance between moon and earth is 3, 84,400 Km. But it is really difficult to remember such a bigger distance value.

So, what can we do? We can try to simplify this bigger value. Why not write 3,84,400 as 620². 620* 620 = 3, 84,400.

Thus, the exponent of a number implies about the number of times to use the number in a multiplication.

What are Exponents?

A simple natural number when expressed in the form of power, then it is called an exponent or power or indices.

In other words, we can also say that exponents are expressed in the form of multiple of the simplest numbers. Thus, by the exponent, you can simplify or reduce a bigger number into a much smaller number.

Exponents Examples

For Example, 625 can be written as 25² or 25 * 25 = 625. Here, 25 is the base and 2 is the exponent. Exponents can also be termed as power. And thus you can also say 2 is both exponent and power.

Exponent is extremely useful in regular mathematics. When you have large digits, then it is not possible to remember them fully. And thus they are expressed in the form of exponents.

For Example, 5612161 is such a large number. You can simplify and write it as, 2369 * 2369 = 5612161.

You can also write it as, 2369² = 5612161.

Types of Exponents

There are several types of exponents on the basis of their power. Let’s have a look at some of the major types of exponents:

1. Positive Exponent – These exponents are the ones which have positive numbers as their power. For Example, 49 = 7² is a positive exponent.

2. Negative Exponent – These exponents are the ones which have negative numbers as their power. For Example, 8^-2. This needs to be simplified.

Let’s understand how to simplify a negative exponent.

When you have the number 8^-2.

You need to first add 1 in the numerator and convert the number in question into the denominator. For Example, 8-2 = -1/8 * 1/8 = 1/64.

Thus, 8^-2 can also be written as 1/64.

3. Zero Exponent – These exponents are the ones when you have 0 in the power. 0 is equal to 1 and thus you don’t need to perform any special operation here.

For Example, 90 is equal to 91. It can also be written as,

9⁰ = 9¹ = 9.

4. Rational Exponent – These exponents are the ones which have a rational number or fractions as the power. For Example, 10^1/2 is the example of a rational exponent.

10^1/2 can be called as 2 roots of 10.

Rule of Exponents

Following are the simple but extremely useful rules of exponents:

• A⁰ = 1: As per this rule, the power of a natural number is 0, and then the result will be 1.

• A^mAⁿ = A^m+n: As per this rule, when there are two same natural nos., then the different powers can be added together.

• A^m/n = A^m-n: As per this rule, when power is in the rational form, then the numerator and denominator can be subtracted and brought to simple terms.

• (A^m)ⁿ = A^m.n: As per this rule, when there are brackets between two different powers then it can be multiplied and then solved accordingly.

The above rules will help in solving any of the mathematical equations.

For Example,

(5²)³ = 5^2*3 = 5⁶ = 5 * 5 * 5 * 5 * 5 * 5 = 15625

Let’s solve one more example,

Simplify, 64²/2²

= (64/2)²

= (32)²

= 1024

Relation between Positive and Negative Power

a^x = 1/a^-x

Where, a is the base. X is the power or exponent.

Let’s understand this with an example,

5² = 1/5^-2

Solved Examples:

Example: Simplify 6² + 4³ + 7² - 2³

Solution:

= 6 * 6 + 4 * 4 * 4 + 7 * 7 – 2 * 2 * 2

= 36 + 64 + 49 – 8

= 149 – 8

= 141

Example: Solve 16/2² + 32/4²

Solution:

= 16/2*2 + 32/4*4

= 16/4 + 32/16

= 4 + 2

= 6

Example: Solve 10² * 10³ + 2⁵

Solution:

= 100 * 1000 + 32

= 100000 + 32

= 100032

Example: Solve 17² * 18³

Solution:

= 289 * 5832

= 1685448

Fun Facts

• Exponents, on contrary to multiplication, do NOT "distribute" over addition.

• Anything raised to the power zero (0) is just "1" (until and unless "anything" is not itself zero).