Exponent represents repeated multiplication of a number by itself. For example:
\(2^3\), here 2 is called the ‘base’ and ‘3’ is the exponent which indicates how many times the base is used as a factor. We can also say ‘2 raised to the power of 3’. Base with power 2 and 3 have special names. Like \(2^2\) is ‘2 square’ and \(2^3\) is ‘2 cube’.
\(2^3 = 2 \times 2 \times 2\)
\(5^2 = 5 \times 5\)
\(6^4 = 6 \times 6 \times 6 \times 6\)
\(x^4 = x \times x \times x \times x\) , here x is a variable with exponent 4
Rules to follow in solving expression involving exponents:
Rule 1: \(a^\frac{1}{n} = \sqrt[n]{a} \) (where a is any non-negative real number)
\(27^\frac{1}{3} = \sqrt[3]{27}\)
\(64^\frac{1}{2} = \sqrt{64} = 8\)
Rule 2: \(a^m \times a^n = a^{m+n} \) (a is a non-zero real number and m, n are integers)
\(3^2 \times 3^1 = 3^{2+1} = 3^3 = 27\)
\(5^2 \times 5^2 = 5^{2+2} = 5^4 = 625\)
Rule 3: \(a^m \div a^n = a^{m-n} \) (a is a non-zero real number and m, n are integers)
\(3^5 \div 3^2 = 3^{5-2} = 3^3 = 27\)
Rule 4: \((a^m)^n = a^{mn} \) (a is a non-zero real number and m, n are integers)
\((2^2)^3 = 2^{2\times3} = 2^6 = 64\)
Rule 5: \((a\times b)^m = a^m \times b^m\) (a is a non-zero real number and m is integer)
\((2\times3)^2 = 2^2 \times 3^2=36\)
Rule 6: \((\frac{a}{b})^m = a^m \div b^m\) (a, b: non-zero real number and m: integer)
\((\frac{12}{3})^2 = 12^2 \div 3^2 = 16\)
Applying the above rules below also holds true:
\(a^0 = 1\)
Example:
\(2^0 = 1 \text{ , } 6^0 = 1\)
\(a^{-m} = \frac{1}{a^m}\)
Example:
\(2^{-3} = \frac{1}{2^3}\)
