The index laws

There are five useful laws for handling indices with the same base. They are called the index laws.

Index law 1: A rule for multiplying two or more powers of the same base

Example 1

\(7^5 × 7^3 = (7 × 7 × 7 × 7 × 7) × (7 × 7 × 7) = 7^8\)

There are 5 + 3 = 8 factors of 7 in the product, so the product is 7\(^8\).

We add the two indices to get the result:

\(7^5 × 7^3 = 7^8\).

Example 2

Write 5\(^2\) × 5\(^3\) as a single power of 5.

Solution

\(5^2 × 5^3 = 5^{2+3} = 5^5\)

Index law 2: A rule for dividing two or more powers of the same base

Example 3

\(7^6 ÷ 7^3\)	\(= \dfrac{7\hspace{-3.5mm}\setminus × 7\hspace{-3.5mm}\setminus × 7\hspace{-3.5mm}\setminus × 7 × 7 × 7}{7\hspace{-3.5mm}\setminus × 7\hspace{-3.5mm}\setminus × 7\hspace{-3.5mm}\setminus}\)	(three factors cancel out)
	= 7 × 7 × 7	(this leaves 6 – 3 = 3 factors behind)
	= 7^{6–3}	(notice that we subtract the indices to get the result)
	= 7^3

Example 4

Write \(5^6 ÷ 5^2\) as a single power of 5.

Solution

\begin{align}5^6 ÷ 5^2 &= 5^{6-2}\\ &= 5^4\end{align}