# Exponential Terms

When you raise a number or an expression to a power (which itself might be another number or an expression), you have an **exponential term**. Some examples of exponential terms:

\[{2^3},\;{\pi ^7},\;{2^{\left( {\frac{1}{2}} \right)}},\;{10^{\left( {\frac{1}{{\sqrt 2 }}} \right)}},\;{2^x},\;{x^{10}},\;{x^y}\]

In any exponential term, we have a **base**, and an **exponent** – which is the power to which the base is raised. It is easy to interpret exponential terms where the exponent is a natural number. For example,

\[\begin{array}{l}{2^3} = 2 \times 2 \times 2\\{3^7} = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3\\{\left( {\sqrt 2 } \right)^4} = \sqrt 2 \times \sqrt 2 \times \sqrt 2 \times \sqrt 2 \\{\left( {ab} \right)^3} = ab \times ab \times ab\end{array}\]

Thus, in these cases, you simply multiply the base by itself *n* number of times, where *n* is your exponent. But what if *n* is a negative integer? In that case, the exponentiation operation will work as follows:

\[\begin{align} &{2^{ - 3}} = {( {\frac{1}{2}} )^3} = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\\& {( {\sqrt 2 } )^{ - 4}} = {( {\frac{1}{{\sqrt 2 }}} )^4} = \frac{1}{{\sqrt 2 }} \times \frac{1}{{\sqrt 2 }} \times \frac{1}{{\sqrt 2 }} \times \frac{1}{{\sqrt 2 }}\\ & {( {ab} )^{ - 3}} = \frac{1}{{ab}} \times \frac{1}{{ab}} \times \frac{1}{{ab}}\end{align}\]

What if the exponent is 0? Any base raised to the power 0 will be 1. It is easy to prove this, and we will do so soon. We will also see how to interpret exponential terms where the exponent is not an integer at all – it is a rational or even an irrational number.