Expressions or numbers that involve exponents can be simplified using these laws:

$\begin{align*}a^m \times a^n &= a^{m+n} \\[10pt] \frac{a^m}{a^n} &= a^{m - n} \\[10pt] (a^m)^n&=a^{mn} \\[10pt] a^1&=a \\[10pt] a^0&=1 \\[10pt] a^{-n}&=\frac{1}{a^n} \\[10pt] (a\cdot b)^n&=a^n \times b^n \\[10pt] \Big(\frac{a}{b}\Big)^n&=\frac{a^n}{b^n}\end{align*}$

Mathantics summary:

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 $m$ , $n$ ,

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

More generally,

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

The Quotient Law

When a fraction has a common factor on numerator and denominator, it can be simplified. For any number $a$ ,

$\frac{a}{a}=1$

For example:

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

More briefly:

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

Simplifying fractions leads us to the quotient rule.

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

We started with the product of 5 sevens, 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,

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

The Power Law

The power law tells us what do when we have a value expressed with an exponent raised to another exponent:

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

In short,

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

Exponent Zero

The exponent zero can be considered as the number 1 multiplied by any base no times.

An integer exponent records the number of times the number 1 is multiplied or divided by the same number, known as the base.

Move the exponent slider to zero for different bases:

applet link

The number 1 is implied when we are multiplying. 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’.

Starting at any exponent of $7$ we can reach all other exponents either by repeated multiplying by 7 or repeated dividing by 7. We can continue multiplying or dividing indefinitely. We can think of the number 1 as being in the middle of it all. 1 is the number 1 multiplied or divided by 7 (or anything) no times.

Here is the pattern:

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

Another way to consider the exponent zero is with an example like $7^3 \div 7^3$ . 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…).

Negative Exponents

Multiplying by the base once results in adding one to the exponent:

$7^4\times 7 = 7^{4+1}=7^5$

Dividing by the base once results in subtracting one from the exponent:

$7^4 \div 7 = 7^{4-1}=7^3$

If we keep subtracting from the exponent, eventually we will reach a negative exponent:

$7^4 \div 7^2 = 7^{4-2}=7^2=49$

$7^4 \div 7^3 = 7^{4-3}=7^1=7$

$7^4 \div 7^4= 7^{4-4}=7^0=1$

$7^4 \div 7^5= 7^{4-5}= 7^{-1}=\frac{1}{7}$

$7^4 \div 7^6= 7^{4-6}= 7^{-2}=\frac{1}{7^2}$

Note in the last line,

$7^4 \div 7^6 =7^{-2}$

Also,

$\frac{7^4}{7^6}=\frac{\cancel{7}\times \cancel{7} \times \cancel{7}\times \cancel{7}}{\cancel{7}\times \cancel{7} \times \cancel{7}\times \cancel{7}\times 7 \times 7}= \frac{1}{7^2}$

In other words:

$7^{-2}=\frac{1}{7^2}$

In general, a negative exponent is an instruction to divide.

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

The set of calculations above can be written as follows:

$7^4 \div 7^2 = 7^4\times 7^{-2}=7^{4-2}=7^2=49$

$7^4 \div 7^3 = 7^4 \times 7^{-3}=7^{4-3}=7^1=7$

etc

When can a value expressed with an exponent be a negative? (Less than zero)

A negative exponent on a positive base does not result in a negative number.

Example:

$3^{-5}=1\div 3\div3\div3\div3\div3 =\frac{1}{3^5}=\frac{1}{243}$

The fraction $\frac{1}{243}$ is a positive number.

On the other hand, if the base is negative and the exponent is an odd number, the value is negative.

$(-3)^5=(-3)(-3)(-3)(-3)(-3)=(9)(9)(-3)=81(-3)=-243$

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 $2^{0.5} \times 2^{0.5}$ ?