Exponent Laws: Development

Let’s examine 7^n. On this page, our values for n are integers – a negative whole number, zero or a positive whole number.

We have defined the exponent to be a short hand for repeated multiplication.

By the way, we said that n is an integer. It doesn’t have to be. These laws make sense more readily when used with integer exponents.  The exponent can also be rational; irrational; or complex. We use irrational exponents whenever we define an exponential function over a real domain. On this page, we’ll look for the pattern made by integer exponents.

A summary of the laws can be found here.

The Addition Law

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

    \[7 \times (7 \times 7 \times 7)=7^1 \times 7^3 = 7^4\]

or

    \[(7 \times 7) \times (7 \times 7)=7^2 \times 7^2 = 7^4\]

What we notice about the indices 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:

    \[(7 \times 7 \times 7) \times (7 \times 7\times 7\times 7\times 7)=7^8\]

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This leads us to the addition rule, which says: 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 Subtraction 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’. 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 subtraction law.

    \[7^5 \div 7^3=7^{5-3}=7^2\]

or, more generally,

    \[7^m \div 7^n = 7^{m - n}\]

Negative Indices

The subtraction law leads us to negative indices. Suppose we have 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 summary,

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

Or, 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 used, better to say:

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

Exponent Zero

Now suppose we have 7^3 \div 7^3. Well, we already know that anything divided by itself is just 1. How many 343’s are in 343? One!

Let’s see what happens when we apply the subtraction rule:

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

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

Practice

Try this matching exercise to practice:

The Multiplication Law

The multiplication law throws some repeated addition (multiplication) into our repeated multiplication (exponents)! Its quite straightforward:

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

In short,

    \[(7^m)^n=7^{mn}\]

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Distribution over a Product or Division Law

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.

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}.


More Practice


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