# Exponents and Powers

What is Exponent and Power?

Exponents and powers are the two mathematical terms used to simplify the problems, especially in algebra. Hypothetically, both powers and exponents are synonymous, but in mathematical relationships, they are used in different contexts. Power is a core mathematical expression which is used to represent exactly how many times a number should be utilised in a given multiplication. On the other hand, exponents are either positive or negative numbers which represent the power to which the base number is raised. To make it simple to understand, power is an expression that represents repeated multiplication of the same number whereas exponent refers to a quantity that represents the power to which the number is raised.

Exponent: In this expression 104, 10 is the base and 4 is defined as power or exponent+. It is also read as “10 is raised to the power of 4” or simply as “10 to the power 4”.

Power: Power refers to the number of times a number is multiplied by itself. Exponents are defined as the number of times a number is used in the multiplication.

For example, if a number “a” is multiplied by itself for “n” times, the number will be defined as

a X a X a X a X a X……..X a = an  , where an is referred to as “a to the power n”.
Here, a is an integer and n is the natural number.

In the mathematical expression, a is the base and n is the power, thus defining the nth power of a.

Multiplication Law

Multiplication law of exponent states that the multiplication of two expressions with the same base and different powers is equal to the base raised to the sum of the two powers.
For example, if there are two expressions, an and am, then

an X am = an+m , where n and m are the two positive integers.

Division Law

Division law of exponents defines that the division of two expressions with the same base and different powers equals the base raised to the difference of the two powers.

For example, if there are two expressions, an and am, then

an / am = an-m , where n and m are the two positive integers.

Negative Exponent Law

Negative law of exponents states that if a base has a negative power, then the result will be equal to the reciprocal of the base raised to the positive power.

For example, a-m = 1/am

4 Rules for Exponents and Powers

Here we explain the four rules by considering ‘a’ and ‘b’ as two integers while ‘m’ and ‘n’ as the values of the powers.

1. a0  = 1

Any integer with the power zero always equals to 1.
Example. 670  = 1.

1. (am)n = anm

Any integer raised to the power m to the power m, the expression equal to the product of the two powers.

Example. (23)4  = 23X4

1. am  X bm  = abm

When two integers a and b with the same power m are multiplied together, the product will be raised to the power m.

Example, 23 X 43 = (2X4)

1. am/bm = (a/b)m

The division of the two integers a and b with the same power m equals to the division of a and b whole to the power m.

Example, 43 / 23 = (4/2)3

Solved Examples

Example 1: 5 X 5 X 5 = 53 = 125

Example 2: 56 can also be written in a simpler form as 5 X 5 X 5 X 5 X 5 X 5.

Example 3. Simplify 56/5

56/52 = (5 X 5 X 5 X 5 X 5 X 5) / (5X5) = 56-2 = 54 = 625

Did you know?

Calculation obtained by measuring the distance between the Sun and the Earth is 149,600,000 kilometres.

Similarly, the mass of the Sun is calculated and the obtained result is 1,989,000,000,000,000,000,000,000,000,000 kilograms.

These figures are too hard to memorize. Hence exponents and powers method is used to reduce the long-tailed figure to a compact form.

For example, the distance between the Sun and the Earth as stated above is 149,600,000 km. This is written in the simplified form as 1.496 X 10 X 10 X 10 X 10 X 10 X10 X 10 = 1.496 X 108 kms.

Likewise, the mass of the Sun is memorized as 1.989 × 1030 kgs.

1. How to Solve a Question where the Exponent is 1 or 0?

If the exponent or power of an integer is 1, the result will be equal to the integer itself.
For example, 201 = 20

If the exponent or power of an integer is 0, the result will always be equal to 1.
For example, 550 = 1

If the integer and power are both 0, the result is indeterminate.
For example, 00 = either 1 or 0 = indeterminate.

On a side note, ‘^’ symbol is used as a power symbol. For example, you can either write 24 or 2^4. Both the expressions define that 2 is the base and 4 is the power of 2.

2. What is the Difference in a Lesson Plan for Exponents and Powers?

Power and exponent are the two hypothetical terms having almost similar definitions but used in different contexts in a mathematical relationship.

Power is defined as an expression when an integer is multiplied by itself for a number of times. While exponent is defined as the number of times the same number has been used in the multiplication.

Exponent is usually known as power or indices in a mathematical expression. Regardless of their definitions and properties, these two terms are quite interchangeably used in the mathematical world which gives rise to a myriad of confusion. So, each concept should be studied independently.