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$$\textbf{Raising a Product to a Power}$$

Raising a Product to a Power

Let's do one more example.

We are trying to expand $(10x^2)^3.$ If we use the variables from the graphic, this means that $x=10, y=x^2,$ and $a=3.$ Now all we have to do is substitute these values into our formula. Here's a link to the power rule for exponents in case you need it. $$\begin{align} (xy)^a&=x^ay^a\\ \\ (10x^2)^3&=(10)^3(x^2)^3\\ &=1{,}000(x^2)^3\\ &=1{,}000x^{2 \cdot 3} \quad\text{ by the power rule for exponents}\\ &=1{,}000x^{6}\\ \end{align}$$

Different Bases, Same Powers

Sometimes, you will encounter a product of different bases, $x$ and $y,$ that are both raised to the same power, $a.$ With the following property of exponents, you can group these bases together.

Before grouping, all of the powers will have a common factor. You are essentially trying to remove that factor from all of the bases.

Let's work through one more example. This'll get easier as you keep working on them, so hang tight!

$$\begin{alignat*}{2} 8w^3z^6&=2^3w^3z^6 \phantom{10}&&(\text{since $8=2\cdot 2 \cdot 2 = 2^3$})\\ &=2^3w^3(z^2)^3 \phantom{10}&&(\text{by the [power rule for exponents](https://www.expii.com/t/raising-a-power-to-a-power-9169)})\\ &=(2wz^2)^3 \phantom{10}&&(\text{because all the bases are raised to the power of $3$}) \end{alignat*}$$

(Video) Products of Powers

by MathWOEs

Math WOEs has a great video with tons of examples of multiplying powers of the same base.

$\large{\text{Summary}}$

Now, what if you multiply powers of different bases? The quick answer is nothing. $x^3\cdot y^5=x^3y^5$. But if the powers have the same exponent, you can combine them to look like this: $$a^4\cdot b^4=(a\cdot b)^4=(ab)^4$$This is definitely less useful than combining powers of the same base, but you should still be familiar and comfortable with this concept.

Let's look at an example. Turn the following into a product raised to a power, $$4x^2y^6$$ First, we want to find a common power. The $x$ is being raised to $2$. Can we write the other terms as something raised to a power of two. We can change $4$ to $2^2$. $$=2^2x^2y^6$$ Similarly, we can turn $y^6$ into $y^{3\times2}$. $$=2^2x^2y^{3\times2}$$ Since all these have a power of $2$, we can pull it out. $$=(2xy^3)^2$$

If you are in doubt of what powers to pull out, there's a trick. Try to find the prime factorization of each exponent. The factors shared in common can be pulled out. If there are multiple such prime number factors this might take a while. However it will ensure you don't miss any powers.