Video: Simplifying Algebraic Expressions Involving Square Roots

Given that π‘₯ = √7/√13 and 𝑦 = √13/√7, find 182(π‘₯ + 𝑦) expressing your answer in simplest form.

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Video Transcript

Given that π‘₯ is equal to root seven divided by root 13 and 𝑦 is equal to root 13 divided by root seven, find 182 multiplied by π‘₯ plus 𝑦, expressing your answer in simplest form.

Our first step here is to rationalize the denominator of both π‘₯ and 𝑦. In order to rationalize the denominator of π‘₯, we multiply the top and bottom of the fraction by root 13. At this point, it is worth remembering two of the laws of surds. Root π‘Ž multiplied by root π‘Ž is equal to π‘Ž, and root π‘Ž multiplied by root 𝑏 is equal to root π‘Žπ‘.

In our question, root seven multiplied by root 13 will be equal to root 91, as seven multiplied by 13 equals 91. Multiplying the denominators gives us 13, as root 13 multiplied by root 13 is equal to 13.

𝑦 was equal to root 13 divided by root seven. To rationalize this surd, we multiply the top and bottom by root seven. Once again, root 13 multiplied by root seven is equal to root 91, and root seven multiplied by root seven is equal to seven.

We now have simplified expressions for π‘₯ and 𝑦. We want to calculate 182 multiplied by π‘₯ plus 𝑦. Let’s first work out what π‘₯ plus 𝑦 would be. π‘₯ plus 𝑦 is equal to root 91 divided by 13 plus root 91 divided by seven. In order to add two fractions, we need to find a common denominator, in this case a number that is in the seven and 13 times table. The lowest common denominator in this case is 91. 13 multiplied by seven is equal to 91.

Multiplying the numerator and denominator of the first fraction by seven gives a seven root 91 divided by 91. Multiplying the numerator and denominator of the second fraction by 13 gives us 13 root 91 divided by 91.

Now that the denominators are the same, we can add the two numerators. Seven root 91 plus 13 root 91 is equal to 20 root 91. Therefore, π‘₯ plus 𝑦 is equal to 20 root 91 divided by 91. We now need to multiply this expression by 182 to get the final answer in this question. 182 divided by 91 is equal to two, as 91 is a half of 182. We’re therefore left with two multiplied by 20 root 91. This is equal to 40 root 91. If π‘₯ is equal to root seven divided by root 13 and 𝑦 is equal to root 13 divided by root seven, then 182 multiplied by π‘₯ plus 𝑦 is equal to 40 root 91.

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