Multiplying and dividing algebraic expressions

In a nutshell

Algebraic terms can be multiplied or divided. If the variables are the same, the laws of indices can be used. When multiplying, different variables can be written together without the multiplication sign between them. For dividing, variables can be written as a fraction. Numbers can be multiplied or divided as usual.

Multiplying

To multiply two terms, multiply numbers as normal and add the powers on the (same) variables using the rule:

$x^a \times x^b = x^{a+b}$

This rule can only be applied if variables are the same. If variables are different, write them together, omitting the multiplication sign. The number on the power tells you how many times that variable has been multiplied together.

Examples

$\begin {aligned}a \times 0 &= 0 \\a \times 1 &= a \\a \times a &= a^2 \\a^2 \times a^3 &= a^5 \\2a^4 \times 3a^5 &= 6a^9 \\ a \times b &= ab \\ 2a^2 \times 5ab &=10a^3b \\\end {aligned}$

Dividing

To divide two terms, divide the numbers as normal and subtract the powers on the (same) variables using the rule:

$x^a \div x^b = x^{a-b}$

This rule can only be applied if the variables are the same. If the variables are different, write them as a fraction.

Examples

$\begin {aligned}\frac {6a} 3 &= 2a \\\\ \frac {12x} {2x} &= 6 \\\\ \frac {15x^2} {3x} &= 5x \\\\ \frac {10a} {2b} &= \frac {5a} b \\\\ \frac {27a^3b^2c} {3a^2c} &= 9ab^2\end {aligned}$