Example: \(5\times5\times5\) can be written as \(5^3\); where 5 is the base and 3 is the exponent or power.
To solve exponent: simply multiply the number by itself, the number of times the power is given:
Example: To solve \(3^4\),multiply the base by itself 4-times\(\left(becausethepoweris4\right)\),as
\(3\times3\times3\times3=81\)
Exponential form: When a number is written in the form of power of its factors, is called exponential form.
To write in exponential form:
Step 1: find all the (prime) factors of the given number.
Step 2: each factor will be the base and count how many times a factor is repeating and that will be the power of that factor.
Example: the number 16 can be written in exponential form as \(2^{4\ }(because\ 2^{4\ }=2\times2\times2\times2=16)\)
Another example: the number 72 can be written as \(72=\ 2\times2\times2\times3\times3\)
The factor 2 is repeating 3 times and factor 3 is repeating 2 times, therefore exponential form will be \(72=2^{3\times}3^2\)
Rules of Exponents: the following are the rules for exponential form of a number:
1) Rule of multiplication: if same ‘bases’ are multiplied with different ‘exponents’; exponents can be added, as \(x^a\times x^b=x^{a+b};\ where\ x,\ a\ and\ b\ can\ be\ any\ numbers.\)
Example: \(2^3\times2^2=2^{3+2}\)
\(\ \ =\ 2^5\)
2) Rule of quotient: if a number divides itself having different exponents then exponents can be subtracted, as
\(\frac{x^a}{x^b}=x^{a-b};\ where\ x,\ a\ and\ b\ can\ be\ any\ numbers.\)
Example: \(\frac{2^3}{2^2}=2^{3-2}\)
\(2^1=2\)
3) Power rule: If a base has a power of power (or double power) then the powers can be multiplied, as
\({{(x}^a)}^b=x^{a\times b};\ where\ x,\ a\ and\ b\ can\ be\ any\ numbers.\)
Example: \({{(2}^3)}^2=2^{3\times2}\)
\(\ \ \ \ =2^6\)
4) Fraction rule: The power of a fraction can be written as the individual power of numerator and denominator, as
\(\left(\frac{x}{y}\right)^a=\frac{x^a}{y^a};\ where\ x,\ y\ and\ a\ can\ be\ any\ numbers.\)
Example: \(\left(\frac{3}{2}\right)^3=\frac{3^3}{2^3}\)
5) Zero exponent: any number having exponent zero (0) is equal to 1, as \(x^0=1;\ wheer\ x\ can\ be\ any\ number.\)
Example: \(2^0=1\)
6) Negative exponent: negative exponent is same as the reciprocal of a positive exponent, as \(x^{-a}=\frac{1}{x^a};\ where\ x\ and\ a\ can\ be\ any\ numbers.\)
Example: \(2^{-3}=\frac{1}{2^3}\)
7) Fractional power: if a number has fractional power then it can be written in the form of root as, \(x^\frac{a}{b}=\sqrt[b]{x^a};\ where\ x,\ a\ and\ b\ can\ be\ any\ numbers.\)
Example: \(8^\frac{2}{3}=\sqrt[3]{8^2} \)
8) Power of a product: the power of a product can be written as the individual power of its factors, as \({(xy)}^a=x^a\times y^a;\ where\ x,\ y\ and\ a\ can\ be\ any\ numbers.\)
Example: \(6^2=(2\times{3)}^2\)
\(\ \ \ \ {=2}^2\times3^2\)