What is an Exponent?
An exponent represents the multiplication of a number n times by itself.
When 3 is multiplied by itself n times, it is written as 3 x 3 x 3 x 3 x 3…..n times = \(3^n\)
The above expression, \(3^n\), is written as 3 multiplied by itself n times. Here, 3 is known as base and n is known as exponent. Exponents are also known as power or indices.
For example:
- 2 × 2 × 2 = \(2^3\)
- \(x~\times~x~\times~x~\times~x~\times~x~\times~\) = \(x^5\)
- a × a × 5 × 5 × 5 = \(a^2 5^3 \)
Laws of Exponent
Following are the laws of exponents:
- Product of powers property
- Power of a power property
- Power of a product property
- Quotient of powers property
- Understanding zero exponents
- Understanding negative exponents
Product of Powers Property -1
When we multiply the power with the same base, we add the exponents of the base to obtain the product.
\(a^m~\times~a^n=a^{m+n}\)
For example:
\(4^3~\times~4^2=4^{3+2}\) Product of Power Property
= \(4^5\) Simplify
Power of a Power Property - 2
When we have to find out the result of the power of a power, then we multiply the exponents of the base to obtain the result.
\((a^m)^n=a^{mn}\)
For example:
\((4^3)^2=4^{3~\times~2}\) Power of a Power Property
= \(4^6\) Simplify
Power of a Product Property - 3
When we have to find out the result of the power of a product of two or more numbers, then we find the power of numbers and multiply them.
\((a~\times~b)^n=a^n~\times~b^n\)
For example:
\((3y)^3=3^3~\times~y^3\) Power of a Product Property
= \(27y^3\) Simplify
Quotient of Powers Property
When we divide the power with the same base, then we get the by subtracting the exponents of the base.
\(\frac{a^m}{a^n}=a^{m-n}\)
For example:
\(\frac{7^6}{7^4}=7^{6-4}\) Quotient of Power Property
= \({7^2}\) Simplify
Understanding Zero Exponents
When the power of any non-zero number is zero, then, its value is equal to 1.
\(a^0=1\)
For example:
\(21^0=1\) Zero Exponent
Understanding Negative Exponents
The negative power of any non-zero number is equal to the reciprocal of the positive power of that number.
\(a^{-n}=\frac{1}{a^n}\)
For example:
\(5^{-4}=\frac{1}{5^4}\) Negative Exponent
= \(\frac{1}{625}\) Simplify
Solved Examples
Example 1: Simplifying the expression.
\(\frac{9^5~\times~9^3}{9^6}\)
Solution:
\(\frac{9^5~\times~9^3}{9^6}\) = \(\frac{9^{5~+~3}}{9^6}\) Product of Powers Property
= \(\frac{9^8}{9^6}\) Simplify
= \(9^{8~-~6}\) Quotient of Power Property
= \({9^2}\) = 81 Simplify
Example 2: Simplifying the expression.
\(\frac{2^{-3}~\times~r^0~\times~y^4}{y^2}\)
Solution:
\(\frac{2^{-3}~\times~r^0~\times~y^4}{y^2}\) = \(\frac{2^{-3}~\times~1~\times~y^4}{y^2}\) Zero Exponent
= \(\frac{2^{-3}~\times~y^4}{y^2}\) Simplify
= \(\frac{y^4}{2^3~\times~y^2}\) Negative Exponent
= \(\frac{y^{4-2}}{2^3}\) Quotient of Power Property
= \(\frac{y^2}{8}\) Simplify
Example 3: John imagines that he will eat 2 chocolates today then double that tomorrow then double that on the day after tomorrow and so on till the 8th day. How many chocolates did he eat on the 8th day?
Solution:
John is imagining that he will eat 2 chocolates today then double that tomorrow, then double that on the day after tomorrow and so on.
So, on the 8th day he will eat a total 8th exponent of 2,
\(2^8\) = 256 chocolates.
An exponent represents the multiplication of a number n times by itself.
When we divide the power with the same base, then in the result we subtract the exponents of the base.
When we find out the negative power(n) of any non-zero number then we get the result as reciprocal of the positive power of n.
When the power of any non-zero number is zero then its value is equal to 1.