Exponents

First glance

When we write 5³, it is said that 5 is raised to the power of three. Here 5 is called the base and 3 is called the exponent. It means 5 is multiplied three times to itself. For example 5³ means 5x5x5. Exponents are also called index. Exponents are extremely useful in expressing very large or very small numbers. Exponents are of two types: Positive exponents and negative exponents. For example in 8² where 2 is a positive integer and in 7^-4 where -4 is a negative integer.

Even fractions can be expressed with the help of exponents such as (7/9)². It means 7 x 7/9 x 9. therefore the result will be 49/81. There are few principles in exponents like:

In depth

In case of positive exponents, 3⁵ means 3 x 3 x 3 x 3 x 3. However, in case of negative exponents 3^-5  means 1/ 3⁵
i.e., 1/3 x 3 x 3 x 3 x 3.

When we need to multiply numbers that have same base then we can just add the exponents. For example we need to multiply 2^3  to  2^6  then one of the ways to get the product is to get the value of  2³  and  2^6   separately and then multiply them to get the final product. However, an easier method and prevalent method is to add the exponents that is 2^3  x 2^6   =  2³⁺⁶  =  2⁹.

If the multiplication consists of more than two different bases such as 2⁴ x 2⁶ x 3³ x 3^5  , then the right process to calculate this is, 2^4+6  x  3^3+5  =  2^10  x 3⁸

When we need to divide two numbers of same bases then the suggested method is to subtract the exponent of divisor from the exponent of the dividend. For example if we divide  7⁸ / 7⁶, then the best way to calculate the quotient is  7^8-6 = 7². However, there can be two different bases in the dividend itself, but those two bases need to remain in case of the divisor too. Fore example,  2⁴ x 3⁵ / 2^2  x 3². Then the answer will be 2^4-2  x  3^5-2  =  2^2  x 3³ .

Reciprocal is a huge factor in case of exponents. For example the value of (3/5)^-3  will be the reciprocal of (3/5)³ that is, it will be (5/3)³.

There are more rules to division such as if the value of the exponent of the dividend is less than the value of the exponent of the divisor then it gives a quotient with a negative exponent. For example,

5² / 5⁵
= 5^2-5
= 5^-3
= 1/5³

Examples:

Multiply:

A.)

5^8  x 5^5
 = 5^8+5
 = 5¹³

B.)

2³ x 3⁴ x 2⁵ x 3⁹
=  2^3+5  x  3⁴⁺⁹
=   2⁸   x 3¹³

Divide:

A.)

6^5  by 6^3
 = 6^{5 -3
Â} = 6²

B.)

7^7  by 7³
=  7^{7 – 3}
=  7⁴

Test your knowledge – try our Exponents Test.