Laws of Indices

Law 1: Multiplication Law

When you multiply similar terms, you need to add their powers.

$a^{textcolor{blue}m} times a^{textcolor{red}n}  = a^{textcolor{blue}m+textcolor{red}n}$ 

The multiplication law applies to all numbers, negative numbers and fractional powers.

Example:

$begin{aligned} a^3 a^4 &= a^{3+4} = a^7 x^4 x^{-1} &= x^{4-1} = x^3 (x+1)^2 (x+1)^3 &= (x+1)^{2+3} = (x+1)^5 t^{frac{1}{5}} t^{frac{2}{5}} &= t^{ frac{1}{5} + frac{2}{5}} = t^{frac{3}{5}} xy^2 cdot x^3 y^{-1} &= x^{1+3} y^{2-1} = x^4 y end{aligned}$

Note: When there are multiple variables, you need to add the powers separately for each variable.

Law 2: Division Law

When you divide similar terms, you need to subtract their powers.

$dfrac{a^{textcolor{blue}m}}{a^{textcolor{red}n}} = a^{textcolor{blue}m} div a^{textcolor{red}n}  = a^{textcolor{blue}m – textcolor{red}n}$ 

The division law applies to all numbers, negative numbers and fractional powers.

Example:

$begin{aligned} dfrac{a^6}{a^4} &= a^{6-4} = a^2 [1.2em] dfrac{x^3}{x^{-1}} &= x^{3-(-1)} = x^4 [1.2em] dfrac{y^2}{y^frac{1}{2}} &= y^{2 – frac{1}{2}} = y^{frac{3}{2}} [1.2em] dfrac{x^2 y^4}{x^3 y} &= x^{2-3} y^{4-1} = x^{-1} y^3 end{aligned}$

Note: When there are multiple variables, you need to subtract the powers separately for each variable.

Law 3: Multiple Powers Law

If you have a power that is raised to another power, then you multiply the powers.

$(a^{textcolor{blue}m})^textcolor{red}n = a^{textcolor{blue}m textcolor{red}n}$ 

The multiple powers law applies to all numbers, negative numbers and fractional powers.

Example:

$begin{aligned} (x^3)^2 &= x^{3 times 2} = x^6 (y^4)^{-2} &= y^{4 times -2} = y^{-8} (x^2y)^3 &= (x^2)^3 y^{1 times 3} = x^{2 times 3} y^3 = x^6 y^3 end{aligned}$

Note: When there are multiple variables, you need to multiply the powers separately for each variable.

Law 4: Power 0 Law

Any number or letter to the power $0 = 1$

$a^{textcolor{red}0} = textcolor{blue}1$

Example:

$begin{aligned} 12^0 &= 1 x^0 &= 1 end{aligned}$

Law 5: Roots as Powers Law

Roots, for example square roots or cube roots, can be written as powers.

$a^{frac{1}{textcolor{blue}m}} = sqrt[textcolor{blue}m]{a}$

Example:

$begin{aligned} 9^{frac{1}{2}} &= sqrt{9} = 3 64^{frac{1}{3}} &= sqrt[3]{64} = 4 x^{frac{1}{4}} &= sqrt[4]{x} end{aligned}$

Law 6: Fractional Powers Law

A power that is represented as a fraction means the power of a root or the root of a power. This extends on law 5.

${a}^{{frac{textcolor{blue}{m}}{textcolor{red}{n}}}} = sqrt[textcolor{red}{n}]{{a}^textcolor{blue}{m}} =(sqrt[textcolor{red}{n}]{{a}})^textcolor{blue}{m}$

Example:

$begin{aligned} 16^{frac{3}{2}} &= (16^{frac{1}{2}})^3 = (sqrt{16})^3 = 4^3 = 64 27^{frac{2}{3}} &= (27^{frac{1}{3}})^2 = (sqrt[3]{27})^2 = 3^2 = 9 end{aligned}$

Law 7: Negative Powers Law

A negative power puts the positive power on the bottom of a fraction.

$a^{-textcolor{blue}m} = dfrac{1}{a^{textcolor{blue}m}}$

Example:

$begin{aligned} 3^{-2} &= dfrac{1}{3^2} = dfrac{1}{9} [1.2em] (3x-1)^{-1} &= dfrac{1}{3x-1} end{aligned}$