2.05 Power of a product property

What if we want to convey that 8\times2 is the base of an exponential expression, and the whole thing is being raised to some power? We can use parentheses to make this more clear, as outlined below.

Let's think about the expression 8\times 8\times 8\times 8\times 2 \times 2 \times 2 \times 2. How can we write this in a more compact form, using our knowledge of exponent laws? Firstly, we can see that there are four groups of 8 and four groups of 2 being multiplied together. We can simplify the two parts separately like so: 8\times 8\times 8\times 8\times 2 \times 2 \times 2 \times 2 = 8^{4} \times 2^{4}.

That is a good start, but there is another way we can approach this. Using the fact that multiplication is commutative (the order that we multiply the numbers doesn't change the result) we can rearrange the product to get 8\times 2\times 8\times 2\times 8\times 2\times 8\times 2.

Notice that this expression has the same number of 8s and 2s, just in a different order. Now if we look at groups of 8\times 2, we can treat each one as a separate base and simplify the expression in the following way.

Using two different approaches, we have seen that 8\times 8\times 8\times 8\times 2 \times 2 \times 2 \times 2 can be written as either 8^{4} \times 2^{4} or (8\times 2)^{4}, and this must mean that 8^{4}\times 2^{4}= (8\times 2)^{4}. In the second case, the base is "8\times 2" and the power is 4.

We can use this specific example to arrive at a general rule about bases that are products of two numbers, or two variables, or a number and a variable.

For the product of any numbers a and b in the base, and for any number n in the power, (ab)^{n}=a^{n}b^{n}

The power of a product rule states that a product raised to a power is equivalent to the product of the two factors each raised to the same power.

Examples