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

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

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.

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

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,

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

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,

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

Exponent Zero

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

An 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. There is no highest or lowest value, but 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.

Written out as a 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 this is to 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…).

Negative Indices

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,

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

Therefore

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

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:

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

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

etc

A negative exponent does not result in a negative number. A negative base raised to an odd integer results in a negative number.

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