What is Exponent?
An exponent represents the number of times a number is multiplied by itself.
For example, if 7 is multiplied by itself for n times, this can be represented as:
7 x 7 x 7 x 7 x 7….. x n times = 7\(^n\)
In the above expression, 7\(^n\), is written as 7 multiplied by itself n times. Here 7 is called base and n is called exponent.
Properties of Exponents
The following properties are used to solve exponents.
- Product of Power Property
When we multiply the power with the same base, we add the exponents of the base.
a\(^m\) . a\(^n\) = a\(^{m~+~n}\)
- Power of a Power Property
When we have to find out the result of the power of a power, we multiply the exponents of the base.
\(\left(a^m\right)^n\) = a\(^{mn}\)
- Power of a Product Property
When we have to find out the result of the power of a product of two or more numbers, we find the power of numbers and multiply them.
(a . b)\(^n\) = a\(^n\) . b\(^n\)
- Quotient of Power Property
When we divide the power with the same base, we subtract the exponents of the base.
\(\frac{a^m}{a^n}\) = \(a^{a+b}\)
- Zero Exponent
When we find that the power is zero of any non zero number, we get the result as 1.
\(a^0\) = 1
- Negative Exponent
When we have a negative exponent of any non zero number, we get the result as the reciprocal of power with a positive exponent.
a\(^{-n}\) = \(\frac{1}{a^n}\)
Formulas of Exponents
- a\(^m\) . a\(^n\) = a\(^{m~+~n}\)
- \(\left(a^m\right)^n\) = a\(^{mn}\)
- (a . b)\(^n\) = a\(^n\) . b\(^n\)
- \(\frac{a^m}{a^n}\) = \(a^{a+b}\)
- \(a^0\) = 1
- a\(^{-n}\) = \(\frac{1}{a^n}\)
Where a is base of power, and m and n are the exponents of power.
Solved Examples
Example 1: Simplify the expression: \(\frac{4^5~.~3^3}{12^3}\).
Solution:
\(\frac{4^5~.~3^3}{12^3}\) [Write the expression]
= \(\frac{4^5~.~3^3}{(4~.~3)^3}\) [Power of Product property ]
= \(\frac{4^5~.~3^3}{4^3~.~3^3}\) [Simplify]
= 4\(^{5-3}\) . 3\(^{3-3}\) [Quotient of Power property ]
= 4\(^2\) . 3\(^0\) [Simplify]
= 4\(^2\) . 1 [Zero exponents]
= 16 [Simplify]
Example 2: Simplifying the expression: \(\frac{7^4~.~(x~.~7)^7~.~x^2}{7^{-3}~.~x^4}\).
Solution:
\(\frac{7^4~.~(x~.~7)^7~.~x^2}{7^{-3}~.~x^4}\) = \(\frac{7^4~.~x^7~.~7^7~.~x^2}{7^{-3}~.~x^4}\) [Power of Product property]
= \(\frac{7^{4+7}~.~x^{7+2}}{7^{-3}~.~x^4}\) [Product of Power property]
= \(\frac{7^{11}~.~x^{9}}{7^{-3}~.~x^4}\) [Simplify]
= \(\frac{7^{11}~.~x^{9-4}}{7^{-3}}\) [Quotient of Power property ]
= \(\frac{7^{11}~.~x^{5}}{7^{-3}}\) [Simplify]
= \(7^{11}~.~x^5~.~7^3\) [Negative exponents]
= \(7^{11~+~3}~.~x^5\) [Product of Power property]
= \(7^{14}~.~x^5\) [Simplify]
Example 3: In a sculptor’s workshop there are multiple sculptures of the same type. The smallest sculpture is 2 cm tall. The height of each successive sculpture in the workshop is twice the height of the previous sculpture. What are the different heights of the figures in the workshop if the height of the tallest sculpture is 256 cm?
Solution:
According to the question and the image, the smallest sculpture is 2 cm and the largest sculpture has a height of 256 cm. Since the height is progressing by multiplying the previous height by two, these are the heights of the sculptures:
1st sculpture: 2 cm
2nd sculpture: 2\(^2\) = 2 x 2 = 4 cm
3rd sculpture: 2\(^3\) = 2 x 2 x 2 = 8 cm
4th sculpture: 2\(^4\) = 2 x 2 x 2 x 2 = 16 cm
5th sculpture: 2\(^5\) = 2 x 2 x 2 x 2 x 2 = 32 cm
6th sculpture: 2\(^6\) = 2 x 2 x 2 x 2 x 2 x 2 = 64 cm
7th sculpture: 2\(^7\) = 2 x 2 x 2 x 2 x 2 x 2 x 2 = 128 cm
8th sculpture: 2\(^8\) = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256 cm
So, the different heights of the sculptures in the workshop are 2 cm, 4 cm, 8 cm, 16 cm, 32 cm, 64 cm, 128 cm and 256 cm. In terms of the powers of 2, the sculptures are ordered as, 2\(^1\) cm, 2\(^2\) cm, 2\(^3\) cm, 2\(^4\) cm, 2\(^5\) cm, 2\(^6\) cm, 2\(^7\) cm, 2\(^8\) cm.
An exponent represents the number of times a number is multiplied by itself.
When we have to find out the result of the power of a product of two or more numbers, we find the power of each number and multiply them.
When we have to find out the result of the power of a power, then we multiply the exponents of the base.
A number when raised to 1 denotes the number itself.