Use the product rule to simplify the expressions in Exercises ...

Question

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. √2x^2⋅√6x

Accepted Answer

Hello, today we are going to be simplifying the given radical expression using the product rule of radicals. So before we jump into this problem, let's quickly review the product rule for radicals. The product rule states that if you have the product of two radicals such as the product of radical A with radical B, then you can take this product by taking the product of the insides of each radical and combining the radicals together. In this case, the product of radical A with radical B will give us radical A times B which is A B. We're going to use this rule to simplify our given expression. The expression given to us is radical four M to the fourth multiplied by radical eight M. Now, in order to simplify this into a single radical, we will need to take the product of the insides of both radicals to do this, we will multiply four and eight together and multiply M to the fourth and M together four multiplied by eight will give us 32 and M to the fourth multiplied by M will leave us with M to the fifth power. Furthermore, this will now all be under a single radical route. So how do we simplify further? Well, recall that the radical symbol allows us to simplify square roots or perfect squares into a single term such as the square root of X squared is just equal to X. What we want to do is you want to break down 32 and empty the fifth into the product of perfect squares. If you're not sure how to break down 32 we can start by drawing a number tree in this number tree, we are going to label two numbers that multiply to give us 32 then break those numbers down further into a simplest form. So two numbers that multiply to give us 32 will be eight and four, four can be rewritten as two multiplied by two. If we look at 88 can be rewritten as four multiplied by two and four can be rewritten as two multiplied by two. So two is the simplest factor of 32 and notice that we have five iterations of two. What this means is that we can take the number 32 and rewrite it as two multiplied by itself five times. But keep in mind that we want to rewrite this as perfect squares since we are going to use the radical to simplify this. So if we group up each multiple of two into groups of two, then we can rewrite the product of 32 as two squared. Multiplied by two squared multiplied by two. If we take a look at the term M to the fifth, we want to repeat this process. So M to the fifth can be rewritten as M squared multiplied by M squared multiplied by M. So if we take both products and put them back into the square root symbol, we will get the square root of two squared multiplied by two squared multiplied by two multiplied by M squared, multiplied by M squared multiplied by M. Now, what we want to do is we want now want to use the square root symbol to remove every perfect square as a single term in front of the square root symbol. So notice that we have two iterations of two that are perfect squares and two iterations of M that are perfect squares. That means we can rewrite these in the front of the square root as two multiplied by two multiplied by M multiplied by M anything that was not a perfect square will be left inside the square root. So this will be multiplied by two multiplied by M which will give us two M. Finally, we will simplify the leading terms. So two multiplied by two will be four and M multiplied by M will give us M squared and this will be multiplied by two M. There is no further way that we can simplify the square roots or any other term. So this is going to be the simplest form of the square root expression. And with that being said, the solution to this problem is going to be c So I hope this video helps you in understanding how to simplify the expression using the product tool for radicals. And I'll go ahead and see you all in the next video.

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. √2x^2⋅√6x

Key Concepts

Product Rule

Square Roots

Nonnegative Real Numbers

Watch next