6.5.1 Multiplying Square Roots
The square root of a product is the product of the square roots. In other words:
if both x and y are at least 0.
We now have a sense of what a square root looks like on a graph. We also know how to solve "basic" equations involving square roots. When we studied exponential relationships, we learned how to solve more advanced equations after we better understood how to combine exponentials. Let's play the same game with radicals and see if we can develop techniques that we can then use to solve equations. The basic ways that we can combine numbers are by adding/subtracting and multiplying/dividing. Since radicals are numbers as well, we should study what happens when we add/subtract or multiply/divide radicals.
Let's start by multiplying square roots and see where this takes us. By doing so, we may gain some insight as to how we can solve an equation such as:
\sqrt{9x}\sqrt{4x}=1
Do we have any old tools that might be able to help us to understand multiplication? Thinking back to when we learned about rational exponents, we can see that
\sqrt{9x}=(9x)^{1/2} and \sqrt{4x} = (4x)^{1/2}
But then we can use the property of exponents:
(xy)^n=x^ny^n
to see that
\sqrt{9x}=(9x)^{1/2}=9^{1/2}x^{1/2} and that \sqrt{4x}=(4x)^{1/2}=4^{1/2}x^{1/2}.
We can then use another property of exponents:
x^mx^n=x^{m+n}
to combine the x values. We then get that
Revisiting our original equation,
\sqrt{9x}\sqrt{4x}=1
We finally see that the left-hand side of the equation is really just 6x while the right hand side is 1 so that the solution is x=1/6.
That seemed like a lot of work for just one example. Is there a general technique or tool that we gained through solving the equation?
\sqrt{xy}=
\sqrt{xy}=(xy)^{1/2}=x^{1/2}y^{1/2}=\sqrt{x}\sqrt{y}
And this tells us that:
\sqrt{xy}=\sqrt{x}\sqrt{y}
with the caveat that we can only apply this rule if x and y are both at least 0, or we'd be taking the square root of a negative number.
Now that we have this formula, we can use it to better understand radicals with large numbers under the radical sign. For example, it might be tough to get a handle on the size of \sqrt{200}. However, using our new understanding of how products work, we can take advantage of any perfect squares that divide into 200 to simplify the radical. In other words,
\sqrt{200}=\sqrt{100\times 2}=\sqrt{100}\sqrt{2}=10\sqrt{2}
Viewing the radical this way makes it easier to get a better sense of the size of the number. However, one must realize that since we can just plug in the square root of 200 into our calculators, that from a computational standpoint, this method of "simplifying" radicals is more of a passing interest than being a useful technique. Perhaps, however, we may find use for it in a future section.
Explore!
\sqrt{500}=10\sqrt{5}
\sqrt{72}=9\sqrt{2}
Now that we know how to multiply square roots, what's the next natural question?
What about nth roots? Do they behave in a similar manner when multiplied? We'll study this in the next section.
