Rationalize each denominator. Assume all variables represent n...

Question

Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. (1 + √3) / (3√5 + 2√3)

Accepted Answer

Hello, today we are going to be rationalizing the denominator of the following expression. The expression given to us is three plus the square root of five divided by two, multiplied by the square root of 17 plus five, multiplied by the square root of two. So what it means to rationalize the denominator is we want to remove all the square root terms from the denominator. The biggest tool that will help us here is by multiplying by the conjugate of the denominator. Now, if you're unsure what the conjugate of the denominator is, we'll take the quantity that's in the denominator and multiply it by its opposite. So if we have two square root of 17 plus five square root of two in the denominator, then we will want to multiply this quantity by two square root of 17 minus five square root of two. This will allow us to eliminate the square roots in the denominator. However, if we will multiply the denominator by the conjugate, we must also multiply the numerator by the conjugate. So we will multiply the numerator by two square root of 17 minus five square root of two. So now what we need to do is we need to multiply the numerators and denominators together. Let's go ahead and start by multiplying the numerator together. In the numerator, we have the quantity three plus the square root of five, multiplied by two square root of 17 minus five square root of two. In order to simplify this quantity, we will have to foil the two quantities together. In order to foil, we will take the first term of the first group which is three and multiply this to two square root of 17, we will multiply 3 to 2 square root of 17 and that will give us six square root of 17. Continuing this process will take three and multiply this to five square root of two, three, multiplied by negative five square root of two will give us negative 15 square root of two. Now we will take positive square root of five and multiply this to two square root of 17. In order to multiply these quantities together, we will take the terms inside both of the square roots and multiply them together that will give us positive two square root of 17, multiplied by five, which will give us 85. And finally, we will take square root of five and multiply this to negative five square root of two. The product of these two terms will give us negative five multiplied by the square root of five times two, which is 10. Now what we need to do is simplify. But currently, there is no further way to simplify the de the numerator. So we will leave it as is now, what we need to do is we need to simplify the denominator. In the denominator, we have the quantities two square root of 17 plus five square root of two, multiplied by two square root of 17 minus five square root of two, just like the numerator. We will need to foil these quantities together. In order to simplify the denominator, we'll start by multiplying two square root of 17 with two square root of 17. The product of these two terms will give us four multiplied by the square root of 17 squared, two square root of 17, multiplied by negative five square root of two will give us negative 10, the square root of 17 multiplied by two. Now we will take five square root of two and multiply it to two square root of 17. This will give us positive 10 times the square root of 17 multiplied by two. And finally, we will multiply five square root of two with negative five square root of two that will give us negative 25 multiplied by the square root of two squared. So now what we need to do is we need to simplify some of the terms in the denominator. Notice that the middle terms are opposites of each other. Native 10 multiplied by the square root of 17 times two will cancel out when adding positive 10 multiplied by the square root of 17 times two. So we can eliminate these two terms. The square root of 17 squared will simplify to leave us with 17 and we will be left with four multiplied by 17 and finally negative 25 multiplied by the square root of two squared. While the square root of two squared will simplify to give us two, leaving us with negative 25 multiplied by two, four, multiplied by 17 will give us 68 and negative 25 multiplied by two will give us negative 50. Finally, 68 minus 50 will give us the value of 18. And this is going to be the denominator in its simplest form. Now that we have the numerator and denominator. In its simplest form, we are going to rewrite and put together the entire expression. So we simplified the numerator to be six square root of 17 minus 15 square root of two plus two square root of 85 minus five square root of 10. And this will be divided by the simplest denominator which in this case is going to be 18. Finally, what we can do is we can reorder the numerator to have all the positive terms in front and put all the negative terms last. So we can rewrite the final expression to be six square root of 17 plus two square root of 85 minus 15 square root of two minus five square root of 10. And this will be divided by 18. There is going to be no further way to simplify the given expression. So this is going to be the simplest form of the expression with a rationalized denominator. And with that being said, the answer to this problem is going to be a. So I hope this video helped you in understanding on how to rationalize a denominator. And I'll go ahead and see you all in the next video.

Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. (1 + √3) / (3√5 + 2√3)

Key Concepts

Rationalizing the Denominator

Conjugates of Binomials

Properties of Square Roots and Nonnegative Variables

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