Exponents

Definition

 

An exponent says how many times a number (the base) is to be used in a product. In the figure on the right, a is the “base” and b is the “exponent.”

Example: \(\begin{align}3^2\end{align}\)

In this case, 3 is the base, and the exponent of 3 is 2, so 3 is used in the product 2 times. This means the same thing as \(\begin{align}3 \times 3\end{align}\).  Thus,

\(\begin{align}3^2 = 3 \times 3\end{align}\)

which is equal to 9.

Another example: \(\begin{align}2 \times 3^2 \times 5^3\end{align}\)

This is equal to

\(\begin{align}2 \times 3 \times 3 \times 5 \times 5 \times 5\end{align}\)

That is,

the exponent of 2 is 1, so 2 is in the product once
the exponent of 3 is 2, so 3 is in the product twice
the exponent of 5 is 3, so 5 is in the product 3 times

Multiplying numbers with the same base

Example: \(\begin{align}3^2 \times 3^3\end{align}\)

In this case, the base is 3, and the first time its exponent is 2, and the second time its exponent is 3. This can be expressed as:

\(\begin{align}(3 \times 3) \times (3 \times 3 \times 3) = \end{align}\)
\(\begin{align}3 \times 3 \times 3 \times 3 \times 3\end{align}\)

In other words, 3 is a factor in the product 5 times. So the original expression

\(\begin{align}3^2 \times 3^3\end{align}\) is equal to \(\begin{align}3^5\end{align}\)

It’s easy to see that if you multiply two numbers with the same base, then the result will be the base with the two exponents added together:

\(\begin{align}3^2 \times 3^3 = 3^{2+3} = 3^5\end{align}\)

In general, if the base is a, and b and c are exponents, then:

\(\begin{align}a^b \times a^c = a^{b+c}\end{align}\)

Dividing numbers with the same base

Example: \(\begin{align}\frac {5^4}{5^3}\end{align}\)

In this case, the base is 5, and its exponent in the numerator is 4, while its exponent in the denominator is 3. This can be expressed as:

\(\begin{align}\frac{5 \times 5 \times 5 \times 5}{ 5 \times 5 \times 5}\end{align}\)

Notice that the 3 instances of 5 in the denominator cancel out with 3 of the instances of 5 in the numerator, leaving just one 5 in the numerator. In other words:

\(\begin{align}\frac{5 \times 5 \times 5 \times 5}{ 5 \times 5 \times 5} = 5^1 = 5\end{align}\)

It’s obvious that this will always work for fractions of numbers having the same base, so we can make the following rule:

If the base is a, and b and c are exponents, then:

\(\begin{align}\frac{a^b}{a^c} = a^{b-c}\end{align}\)

Or, as in the example above: \(\begin{align}\frac{5^4}{5^3} = 5^{4-3} = 5^1 = 5\end{align}\)

Negative exponents

What happens if we’re dividing numbers with the same base, and the exponent in the denominator is larger than the exponent in the numerator?

Example: \(\begin{align}\frac {5^2}{5^4}\end{align}\)

Then the result is \(\begin{align}5^{-2}\end{align}\)

But if we didn’t subtract exponents, but instead just canceled out 5’s from the numerator and denominator we would get

\(\begin{align}\frac {5^2}{5^4} = \frac{1}{5^2}\end{align}\)

So, we conclude that \(\begin{align}5^{-2} = \frac{1}{5^2}\end{align}\)

In general, if the base is a, and b is an exponent, then:

\(\begin{align}a^{-b} = \frac{1}{a^b}\end{align}\)

Zero exponents

What happens if we’re dividing numbers with the same base, and the exponents in the numerator and denominator are equal?

Example: \(\begin{align}\frac {5^4}{5^4}\end{align}\)

Then the result is \(\begin{align}5^0\end{align}\)

But if we didn’t subtract exponents, but instead just canceled out 5’s from the numerator and denominator we would get

\(\begin{align}\frac {5^4}{5^4} = \frac{1}{1} = 1\end{align}\)

So, we conclude that if the base is a, and the exponent is 0, then:

\(\begin{align}a^0 = 1\end{align}\)

In other words, any number to the 0 power equals 1.

(An exception to this rule is 0 to the 0 power. This is undefined.)

Fractional exponents

Suppose we break an exponent into parts like this:

Example: \(\begin{align}4^1 = 4^{\frac{1}{2} + \frac{1}{2}} = 4^{\frac{1}{2}}\times 4^{\frac{1}{2}}\end{align}\)

But what is \(\begin{align}4^{\frac{1}{2}}\end{align}\) ?

We see from above that

\(\begin{align}4 = 4^1 = 4^{\frac{1}{2}}\times 4^{\frac{1}{2}}\end{align}\)

so

\(\begin{align}4^{\frac{1}{2}}\end{align}\)

must be the square root of 4 (which is 2).

In general, the rule is:

If the base is a, and b and c are integers, then:

\(\begin{align}a^{\frac{b}{c}} = \sqrt[c]{a^b}\end{align}\)

Raising to powers

Example: \(\begin{align}(3^2)^4\end{align}\)

This expands to

\(\begin{align}3^2 \times 3^2 \times 3^2 \times 3^2\end{align}\)

and each of these expands to

\(\begin{align}3 \times 3\end{align}\)

so, the final result is

\(\begin{align}3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3\end{align}\)

In other words, 3 appears in the product 8 times.

Therefore, the general rule is if the base is a, and b and c are integers, then:

\(\begin{align}(a^b)^c = a^{b \times c}\end{align}\)

See also Logarithms