Printable: Classroom handout Exponent Laws, Handout Exponent Laws Solutions

Check out ‘Introduction to Exponents‘ before reading this page.

On this page we see how and why expressions that use exponents with the same base can be simplified.

A summary of the laws can be found here.

The Product Law

Now we know that $7 \times 7 \times 7 \times 7 = 7^4$ . Let’s write that in another way:

$7 \times (7 \times 7 \times 7)=7^1 \times 7^3 = 7^4$

$(7 \times 7) \times (7 \times 7)=7^2 \times 7^2 = 7^4$

What we notice about the exponents is that they add up to four.

Let’s take a second example: suppose we have $7^3 \times 7^5$ .

Writing that out in full give us:

$(7 \times 7 \times 7) \times (7 \times 7\times 7\times 7\times 7)=7^8$

This leads us to the law for the product of two exponents with the same base: for any values $m$ , $n$ ,

$7^m \times 7^n = 7^{m+n}$

Of course, the base doesn’t have to be 7; 7 is used here just for illustration.

The Quotient Law

Do you remember that a fraction is just a division? For example:

$\frac{15}{4}= 15 \div 4 = 3.75$

We also know that when a fraction has a common factor on numerator and denominator, it can be simplified. For example:

$\frac{18}{21}=\frac{6\times 3}{7 \times 3}=\frac{6}{7}\times\frac{3}{3}=\frac{6}{7}\times 1 = \frac{6}{7}$

Usually, we don’t write out all of those steps. You might be more familiar with ‘scoring out’ or ‘cancelling’. Showing the steps without scoring out shows why scoring out works mathematically.

Let’s see how simplifying fractions helps with exponents. This time, let’s use ‘scoring out’.

$7^5 \div 7^3 = \frac{7^5}{7^3}=\frac{7\times 7\times 7\times 7 \times7}{7\times 7\times 7}=\frac{7 \times 7 \times \cancel{7} \times \cancel{7} \times \cancel{7}}{\cancel{7} \times \cancel{7} \times \cancel{7}}=\frac{7^2}{1}=7^2$

Or, in otherwords, we see a common factor of $7\times7\times7$ on the numerator and the denominator. How many 7’s did we ‘score out’? We started with 5, we scored out 3, and we were left with 2. This leads us to the law for the quotient of two powers with the same base:

$7^5 \div 7^3=7^{5-3}=7^2$

or, more generally,

$7^m \div 7^n = 7^{m - n}$

Negative Indices

Dividing cancels out the effect of multiplying.

$7\times 7 \times 7 \div 7 \div 7 \div 7 = 1$

Since a negative number cancels a positive number, we can write division with a negative exponent.

$7^3 \times 7^{-3}= 1$

What if we divide by more 7’s than we multiply with? For example, $7^3 \div 7^5$ .

$7^3 \div 7^5 = \frac{7^3}{7^5}=\frac{7\times 7\times 7}{7\times 7\times 7\times 7 \times 7}=\frac{\cancel{7} \times \cancel{7} \times \cancel{7}}{7 \times 7 \times \cancel{7} \times \cancel{7} \times \cancel{7}}=\frac{1}{7^2}$

In shorter form,

$7^3 \div 7^5 = \frac{7^3}{7^5}= \frac{1}{7^2}=7^{-2}$

It is identical to say

$7^3 \times 7^{-5} = \frac{7^3}{7^5}= \frac{1}{7^2}=7^{-2}$

In general:

$7^{-n}=\frac{1}{7^n}$

When we say $7^3$ we mean, $1\cdot 7^3 = 1 \times 7 \times 7 \times 7 = 343$ . When we say $7^{-3}$ we mean one divided by $7$ three times over, that is:

$7^{-3}=1\cdot 7^{-3}=1 \div 7 \div 7 \div 7$

which is really awkward notation and is not often used. Better to say:

$7^{-3}=1\cdot 7^{-3}=1 \div 7^3=\frac{1}{7^3}$

Exponent Zero

The number 1 is implied when we are multiplying. We can simply say $'7'$ to mean $'1 \times 7'$ or $'7^1'$ . Note that its awkward to say ‘7 times by itself once’. We mean, ‘1 multiplied by 7 once’. Similarly, $7^2$ means ‘1 multiplied by 7 twice’.

It is necessary to state ‘1’ when dividing. To divide by 7 once, we have to state that we are applying ‘divide’ to the number $1$ .

The number $1$ is what we have when we are neither multiplying by 7 nor dividing by 7. This is $7^0$ . Here is the pattern:

$\begin{align*}1\times 7 \times 7 \times 7&=7^3\\[10pt]1\times 7 \times 7 &=7^2\\[10pt]1\times 7&=7^1\\[10pt]1&=7^0\\[10pt]1\div 7&=7^{-1} \\[10pt]1\div 7\div 7&=7^{-2}\\[10pt]1\div7 \div7\div7 &=7^{-3} \end{align*}$

Now suppose we have $7^3 \div 7^3$ . Well, we already know that anything divided by itself is just 1.

Let’s see what happens when we apply the quotient law:

$1=\frac{7^3}{7^3}= 7^3 \div 7^3 = 7^{3-3}=7^0$

In general, $a^0=1$ (with the curious exception of a=0, but only sometimes…).

The Power Law

The power law tells us what do when we have a power of a power:

$(7^3)^5=7^3 \times 7^3 \times 7^3 \times 7^3 \times 7^3 = 7^{3+3+3+3+3}=7^{3 \times 5}=7^{15}$

In short,

$(7^m)^n=7^{mn}$

Distribution over a Product or Quotient

Now suppose we have the number $10^2$ , we know the answer is 100. Let’s write the number 10 as a product and then square it:

$(5 \times 2)^2=(5\times2) \times (5 \times 2)= 5 \times 2 \times 5 \times 2 = 5\times 5 \times 2 \times 2 = 5^2 \times 2^2 = 25 \times 4 = 100$

In other words,

$(5 \times 2)^2= 5^2 \times 2^2$

This example illustrates that, in general:

$(a\cdot b)^n=a^n \times b^n$

Similarly, with division:

$\Big(\frac{10}{2}\Big)^2=\Big(\frac{10}{2}\Big)\times \Big(\frac{10}{2}\Big)=\frac{10^2}{2^2}$

In general,

$\Big(\frac{a}{b}\Big)^n=\frac{a^n}{b^n}$

Take Care though – an exponent does not distribute over addition/subtraction!

Example:

$7^2=49$

$\big(3+4\big)^2=(3+4)(3+4)=7 \times 7 =49 \checkmark$

The common mistake would be to say that $(3+4)^2$ is equal to $3^2+4^2$ but lets figure that out: $3^2+4^2=9+16=25$ which is not 49, so that is simply wrong.

Practice

Applet link

Summary

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Number Sense Question:

What might the value of $2^{0.5}$ be? What meaning does it have? If the math on this page extends to exponents that are not integers, what is the value of $2^{0.5} \times 2^{0.5}$ ?

What about $2^{\pi}$ ? Does it have any meaning?

See if your calculator can make sense of $2^{0.5}$ and $2^{\pi}$ .

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