Index laws

Index laws are the rules used in algebra to simplify expressions involving powers or indices.

Notation In this example: $x^{y}$ x is the base and y is the exponent, index or exponent.

Basic Laws

If you multiply two powers with the same base, you add the indices:

a^{x} \cdot a^{y} = a^{x + y}

x^{3} \cdot x^{2} = x^{3 + 2} = x^{5}

This is similar to multiplying powers but in this case we subtract the indices:

\frac{a ^{x}}{a ^{y}} = a^{x - y}

\frac{x ^{5}}{x ^{2}} = x^{5 - 2} = x^{3}

When you raise a power to another power e.g $(a^{x})^{y}$ , you multiply the indices:

(a^{x})^{y} = a^{x \cdot y}

(x^{2})^{3} = x^{2 \cdot 3} = x^{6}

When you raise a product to a power, you raise each factor to that power:

(a \cdot b)^{x} = a^{x} \cdot b^{x}

(a \cdot b)^{3} = a^{3} \cdot b^{3}

(a \cdot b)^{3} = a^{3} \cdot b^{3} \frac{a ^{3} \cdot a ^{2}}{a ^{4}} = \frac{a ^{3 + 2}}{a ^{4}} = \frac{a ^{5}}{a ^{4}} = a^{5 - 4} = a^{1} = a \frac{( a \cdot b ) ^{2}}{( a ^{2} ) ^{3}} = \frac{( a ^{2} \cdot b ^{2} )}{a ^{6}} = \frac{a ^{2 + 2}}{a ^{6}} = \frac{a ^{4}}{a ^{6}} = a^{4 - 6} = a^{- 2} = \frac{1}{a ^{2}} \frac{6 x ^{4} + 10 x ^{6}}{2 x} = \frac{2 x ^{4} ( 3 + 5 x ^{2} )}{2 x} = x^{3} (3 + 5 x^{2})