We’ll derive the properties of exponents by looking for patterns in several examples. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents.  First, we will look at a few examples that leads to the Product Property.

For example, the notation [latex]5^{4}[/latex] can be expanded and written as [latex]5\cdot5\cdot5\cdot5[/latex], or [latex]625[/latex]. And don’t forget, the exponent only applies to the number immediately to its left, unless there are parentheses.

What happens if you multiply two numbers in exponential form with the same base? Consider the expression [latex]{2}^{3}{2}^{4}[/latex]. Expanding each exponent, this can be rewritten as [latex]\left(2\cdot2\cdot2\right)\left(2\cdot2\cdot2\cdot2\right)[/latex] or [latex]2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2[/latex]. In exponential form, you would write the product as [latex]2^{7}[/latex]. Notice that [latex]7[/latex] is the sum of the original two exponents, [latex]3[/latex] and [latex]4[/latex].

What about [latex]{x}^{2}{x}^{6}[/latex]? This can be written as [latex]\left(x\cdot{x}\right)\left(x\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}\right)=x\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}[/latex] or [latex]x^{8}[/latex]. And, once again, [latex]8[/latex] is the sum of the original two exponents. This concept can be generalized in the following way:

	[latex]{x}^{2}\cdot{x}^{3}[/latex]
What does this mean? How many factors altogether?
So, we have	[latex]{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}={x}^{5}[/latex]
Notice that [latex]5[/latex] is the sum of the exponents, [latex]2[/latex] and [latex]3[/latex].	[latex]{x}^{2}\cdot{x}^{3}[/latex] is [latex]{x}^{2+3}[/latex], or [latex]{x}^{5}[/latex]
We write:	[latex]{x}^{2}\cdot {x}^{3}[/latex] [latex]{x}^{2+3}[/latex] [latex]{x}^{5}[/latex]

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

An example with numbers helps to verify this property.

[latex]\begin{array}{ccc}\hfill {2}^{2}\cdot {2}^{3}& \stackrel{?}{=}& {2}^{2+3}\hfill \\ \hfill 4\cdot 8& \stackrel{?}{=}& {2}^{5}\hfill \\ \hfill 32& =& 32\hfill \end{array}[/latex]

We can extend the Product Property of Exponents to more than two factors.

In the following video we show more examples of how to use the product rule for exponents to simplify expressions.

Simplify Expressions Using the Power Property of Exponents

We will now further expand our capabilities with exponents. We will learn what to do when a term with a power is raised to another power, and what to do when two numbers or variables are multiplied and both are raised to an exponent.  We will also learn what to do when numbers or variables that are divided are raised to a power.  We will begin by raising powers to powers.  See if you can discover a general property.

Let’s simplify [latex]\left(5^{2}\right)^{4}[/latex]. In this case, the base is [latex]5^2[/latex] and the exponent is [latex]4[/latex], so you multiply [latex]5^{2}[/latex] four times: [latex]\left(5^{2}\right)^{4}=5^{2}\cdot5^{2}\cdot5^{2}\cdot5^{2}=5^{8}[/latex] (using the Product Rule—add the exponents).

[latex]\left(5^{2}\right)^{4}[/latex] is a power of a power. It is the fourth power of [latex]5[/latex] to the second power. And we saw above that the answer is [latex]5^{8}[/latex]. Notice that the new exponent is the same as the product of the original exponents: [latex]2\cdot4=8[/latex].

So, [latex]\left(5^{2}\right)^{4}=5^{2\cdot4}=5^{8}[/latex] (which equals 390,625, if you do the multiplication).

Likewise, [latex]\left(x^{4}\right)^{3}=x^{4\cdot3}=x^{12}[/latex]

This leads to another rule for exponents—the Power Rule for Exponents. To simplify a power of a power, you multiply the exponents, keeping the base the same. For example, [latex]\left(2^{3}\right)^{5}=2^{15}[/latex].

	[latex]({x}^{2})^{3}[/latex]
	[latex]{x}^{2}\cdot{x}^{2}\cdot{x}^{2}[/latex]
What does this mean? How many factors altogether?
So, we have	[latex]{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}={x}^{6}[/latex]
Notice that [latex]6[/latex] is the product of the exponents, [latex]2[/latex] and [latex]3[/latex].	[latex]({x}^{2})^{3}[/latex] is [latex]{x}^{2\cdot3}[/latex] or [latex]{x}^{6}[/latex]
We write:	[latex]{\left({x}^{2}\right)}^{3}[/latex] [latex]{x}^{2\cdot 3}[/latex] [latex]{x}^{6}[/latex]

This leads to the Power Property for Exponents.

An example with numbers helps to verify this property.

[latex]\begin{array}{ccc}\hfill {\left({5}^{2}\right)}^{3}& \stackrel{?}{=}& {5}^{2\cdot 3}\hfill \\ \hfill {\left(25\right)}^{3}& \stackrel{?}{=}& {5}^{6}\hfill \\ \hfill 15,625& =& 15,625\hfill \end{array}[/latex]

Watch the following video to see more examples of how to use the power rule for exponents to simplify expressions.

Simplify Expressions Using the Product to a Power Property

We will now look at an expression containing a product that is raised to a power. Look for a pattern.

Simplify this expression.

[latex]\left(2a\right)^{4}=\left(2a\right)\left(2a\right)\left(2a\right)\left(2a\right)=\left(2\cdot2\cdot2\cdot2\right)\left(a\cdot{a}\cdot{a}\cdot{a}\cdot{a}\right)=\left(2^{4}\right)\left(a^{4}\right)=16a^{4}[/latex]

Notice that the exponent is applied to each factor of [latex]2a[/latex]. So, we can eliminate the middle steps.

[latex]\begin{array}{l}\left(2a\right)^{4} = \left(2^{4}\right)\left(a^{4}\right)\text{, applying the }4\text{ to each factor, }2\text{ and }a\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,=16a^{4}\end{array}[/latex]

The product of two or more numbers raised to a power is equal to the product of each number raised to the same power.

The exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents.

An example with numbers helps to verify this property:

[latex]\begin{array}{ccc}\hfill {\left(2\cdot 3\right)}^{2}& \stackrel{?}{=}& {2}^{2}\cdot {3}^{2}\hfill \\ \hfill {6}^{2}& \stackrel{?}{=}& 4\cdot 9\hfill \\ \hfill 36& =& 36\hfill \end{array}[/latex]

If the variable has an exponent with it, use the Power Rule: multiply the exponents.

In the next video we show more examples of how to simplify a product raised to a power.