Exponents (also known as indices or powers) are shorthand for repeated multiplication. The exponent of a number tells us how many copies of it are multiplied together.

For example, $\hspace{0.2em} 3^4 \hspace{0.2em}$ would mean we are multiplying four $\hspace{0.2em} 3$s together. So,

$\begin{align*} 3^4 \hspace{0.2em} &= \hspace{0.2em} 3 \times 3 \times 3 \times 3 \\[1em] &= \hspace{0.2em} 81 \end{align*}$

As you can see, we write the exponent as a superscript (the small number at the top). The number carrying the exponent is known as the base.

A couple more examples of exponents —

- $\hspace{0.2em} 5^2 \hspace{0.2em} = \hspace{0.2em} 5 \times 5 \hspace{0.2em} = \hspace{0.2em} 25 \hspace{0.2em}$
- $\hspace{0.2em} 2^3 \hspace{0.2em} = \hspace{0.2em} 2 \times 2 \times 2 \hspace{0.2em} = \hspace{0.2em} 8 \hspace{0.2em}$

Well, there are a few different ways in which you can read exponents. For example —

$\hspace{0.2em} 3^4 \hspace{0.2em}$ can be read as “three raised to the fourth power”, “three to the fourth power”, or “three to the power of four”. Or more commonly, you could say “three to the fourth” or “three to the four”.

I like the “three to the fourth” format. So, here are a few more examples.

- $\hspace{0.2em} 5^3 \hspace{0.2em}$ — five to the third
- $\hspace{0.2em} 2^6 \hspace{0.2em}$ — two to the sixth
- $\hspace{0.2em} 9^2 \hspace{0.2em}$ — nine to the second

Exponents seem pretty simple and intuitive when they are positive. But what if the exponent is zero? Or negative?

According to the zero exponent rule, any non-zero number (or expression) raised to the zeroth power equals $\hspace{0.2em} 1 \hspace{0.2em}$.

For example —

- $\hspace{0.2em} 5^0 \hspace{0.2em} = \hspace{0.2em} 1 \hspace{0.2em}$
- $\hspace{0.2em} 1.3^0 \hspace{0.2em} = \hspace{0.2em} 1 \hspace{0.2em}$
- $\hspace{0.2em} 9084^0 \hspace{0.2em} = \hspace{0.2em} 1 \hspace{0.2em}$

You could also have algebraic expression as the base, as long as the expression is itself not equal to zero.

For example, $\hspace{0.2em} (y - 3)^0 = 1 \hspace{0.2em}$, as long as $\hspace{0.2em} y - 3 \neq 0 \hspace{0.2em}$. Because remember, for the value to be $\hspace{0.2em} 1 \hspace{0.2em}$, the base cannot be zero.

Similarly, $\hspace{0.2em} (2a + 5b)^0 = 1 \hspace{0.2em}$, with the condition that $\hspace{0.2em} 2a + 5b \neq 0 \hspace{0.2em}$.

There's some disagreement as to what the value of $\hspace{0.2em} 0^0 \hspace{0.2em}$ should be.

Most seem to hold the opinion it should be equal to 1 following the pattern of other numbers. While others say $\hspace{0.2em} 0^0 \hspace{0.2em}$ is indeterminate. Some (a tiny minority) even say it should be equal to zero.

If you are just learning about exponents, you will seldom (if ever) see $\hspace{0.2em} 0^0$, so you can move on without worrying about it much.

Negative exponents indicate repeated division (instead of multiplication).

Makes sense, given that division is the opposite (inverse) of multiplication similar to negative being the opposite of positive.

$\begin{align*} 5^{-2} \hspace{0.2em} &= \hspace{0.2em} \left ( \frac{1}{5} \right )^2 \\[1.5em] &= \hspace{0.2em} \frac{1}{5^2} \end{align*}$

See how the sign of the exponent changes from negative to positive (and the other way round), when we take the reciprocal of the base?

In general,

$a^{-n} = \frac{1}{a^n}$

Here are a few examples.

- $3^{-6} = \frac{1}{3^6}$
- $2^{-4} = \frac{1}{2^4}$
- $\frac{1}{9^{-2}} = 9^2$

What does $\hspace{0.2em} -3^2 \hspace{0.2em}$ mean? Is it $\hspace{0.2em} -3 \times -3 = 9 \hspace{0.2em}$? Or $\hspace{0.2em} -(3 \times 3) = -9 \hspace{0.2em}$?

Well, technically, it’s $\hspace{0.2em} -9 \hspace{0.2em}$. The following side-by-side comparison should help explain what I mean.

$\begin{align*} -3^2 \hspace{0.2em} &= \hspace{0.2em} -(3^2) \\[1em] &= \hspace{0.2em} -(3 \times 3) \\[1em] &= \hspace{0.2em} -9 \end{align*}$

$\begin{align*} (-3)^2 \hspace{0.2em} &= \hspace{0.2em} -3 \times -3 \\[1em] &= \hspace{0.2em} 9 \end{align*}$

Similarly, $\hspace{0.2em} ab^2 \hspace{0.2em}$ is different from $\hspace{0.2em} (ab)^2 \hspace{0.2em}$. The latter would actually be equal to $\hspace{0.2em} a^2b^2 \hspace{0.2em}$.

So yeah, it’s nothing complicated but just be careful so you don’t assume the wrong base.

Exponents can also be fractions. For example, $\hspace{0.2em} 25^{1/3} \hspace{0.2em}$.

So what do fractional exponents mean?

Well, fractional exponents are a way to denote roots. And the denominator of the exponent is the same as the index of the root (or radical). So,

$25 ^ {1/3} \hspace{0.2em} = \hspace{0.2em} \sqrt[3]{25}$

Similarly,

- $32^{1/5} \hspace{0.2em} = \hspace{0.2em} \sqrt[5]{32}$
- $8^{2/3} \hspace{0.2em} = \hspace{0.2em} \sqrt[3]{8^2}$

Note — As you can see in the second example above, only the denominator of a fractional root becomes the index of the root. The numerator retains its position as the exponent.

Now there are two exponents that we frequently come across, $\hspace{0.2em} 2 \hspace{0.2em}$ and $\hspace{0.2em} 3 \hspace{0.2em}$. So much so that it made sense for them to have special names.

The exponent of $\hspace{0.2em} 2 \hspace{0.2em}$ is called a square. So $\hspace{0.2em} 5^2 \hspace{0.2em}$ would be the “square of $\hspace{0.2em} 5 \hspace{0.2em}$” or “$\hspace{0.2em} 5 \hspace{0.2em}$ squared”.

The exponent of $\hspace{0.2em} 2 \hspace{0.2em}$ is called a square. So $\hspace{0.2em} 5^2 \hspace{0.2em}$ would be the “square of $\hspace{0.2em} 5$” or “$5 \hspace{0.2em}$ squared”.

Why square”?

The exponent of $\hspace{0.2em} 2 \hspace{0.2em}$ is gets its name from the fact that raising the side of a square to the power of $\hspace{0.2em} 2 \hspace{0.2em}$ gives us its area.

The exponent of $\hspace{0.2em} 3 \hspace{0.2em}$ is called a cube. So $\hspace{0.2em} 5^3 \hspace{0.2em}$ would be the "cube of $\hspace{0.2em} 5$" or "$5 \hspace{0.2em}$ cubed".

Why “cube”?

The exponent of $\hspace{0.2em} 3 \hspace{0.2em}$ is gets its name from the fact that raising the edge of a cube to the power of $\hspace{0.2em} 3 \hspace{0.2em}$ gives us its volume.

And that brings us to the end of this tutorial on exponents. Until next time.

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