The rules of indices are used to simplify expressions involving powers. These rules apply when multiplying or dividing terms that have the same base.

 

 

Multiplying terms with indices

When multiplying terms with the same base, add the indices.

 

For example:

$$
x^3 \times x^4
$$

 

Add the indices:

$$
x^{3+4}
$$

$$
= x^7
$$

This works because \( x^3 \) means \( x \) multiplied by itself three times and \( x^4 \) means \( x \) multiplied by itself four times, giving seven factors of \( x \) in total.

 

Another example:

$$
2^5 \times 2^2
$$

$$
= 2^{7}
$$

 

 

Multiplying with coefficients

If there are numbers in front of the terms, multiply the numbers separately and then apply the index rule.

 

For example:

$$
3a^2 \times 4a^5
$$

$$
= 12a^{2+5}
$$

$$
= 12a^7
$$

 

 

Dividing terms with indices

When dividing terms with the same base, subtract the indices.

 

For example:

$$
x^7 \div x^3
$$

 

Subtract the indices:

$$
x^{7-3}
$$

$$
= x^4
$$

 

Another example:

$$
5^6 \div 5^2
$$

$$
= 5^{4}
$$

 

 

Dividing with coefficients

Divide the numbers and subtract the indices.

 

For example:

$$
12b^5 \div 3b^2
$$

$$
= 4b^{5-2}
$$

$$
= 4b^3
$$

 

 

Important points to remember

The rules only apply when the bases are the same. For example:

$$
x^2 \times y^3
$$

cannot be simplified using index rules because the bases are different.

 

Using the rules of indices correctly makes multiplying and dividing algebraic terms quicker and helps keep expressions in a simplified form.