Video: Simplifying Numerical Expressions Involving Square Roots

Given that 𝑥 = √12 + √7 and 𝑦 = √192 + √112, express 𝑥 in terms of 𝑦.

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Video Transcript

Given that 𝑥 is equal to root 12 plus root seven and 𝑦 is equal to root 192 plus root 112, express 𝑥 in terms of 𝑦.

In order to answer this question, we firstly need to simplify the surds: root 192 and root 112. In order to simplify a surd, we need to find a square number that is a factor of the number underneath the square root. This factor has to be greater than one.

16 is a factor of 192. Therefore, root 192 can be written as root 16 multiplied by root 12. The square root of 16 is four. Therefore, root 192 is equal to four root 12. This means that the first term in the 𝑦 expression, root 192, is four times the first term in the 𝑥 expression, root 12.

16 is also a factor of 112, as 16 multiplied by seven is equal to 112. Once again, the square root of 16 is equal to four. Therefore, root of 112 can be rewritten as four root seven. We can, therefore, see that the second term in the 𝑦 expression, root 112, is four times the second term in the 𝑥 expression, root seven.

Four root 12 plus four root seven is equal to the root of 192 plus the root of 112. Factorizing out a four on the left-hand side gives us four multiplied by root 12 plus root seven is equal to root 192 plus root 112. Root 12 plus root seven is equal to 𝑥 and root 192 plus root 112 is equal to 𝑦. This means that four 𝑥 is equal to 𝑦.

We were asked to express 𝑥 in terms of 𝑦. This means we need to make 𝑥 the subject. Dividing both sides of the equation by four gives us 𝑥 is equal to 𝑦 divided by four or 𝑦 over four.

If 𝑥 is equal to root 12 plus root seven and 𝑦 is equal to root 192 plus root 112, then 𝑥 is equal to 𝑦 divided by four.

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