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Question Video: Using the Properties of Conjugate Irrational Numbers to Evaluate Algebraic Expressions Mathematics • Second Year of Preparatory School

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Given that π‘₯ = 52/(2√19 βˆ’ 2√6), and 𝑦 is the conjugate of π‘₯, find the value of π‘₯²𝑦².

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Video Transcript

Given that π‘₯ equals 52 over two root 19 minus two root six and 𝑦 is the conjugate of π‘₯, find the value of π‘₯ squared 𝑦 squared.

Recall that the conjugate of an expression of the form π‘Ž root 𝑏 plus 𝑐 root 𝑑 is π‘Ž root 𝑏 minus 𝑐 root 𝑑. Notice that π‘₯ is not currently in this form.

In order to convert π‘₯ into the form π‘Ž root 𝑏 plus 𝑐 root 𝑑, let’s rationalize the denominator. We do this by multiplying the numerator and the denominator by the conjugate of the denominator, that is, by two root 19 plus two root six over two root 19 plus two root six. We have 52 times two root 19 plus two root six over two root 19 minus two root six times two root 19 plus two root six, which we now simplify.

In the numerator, we have 104 root 19 plus 104 root six. And expanding the denominator, two root 19 multiplies with itself to give four times 19. Two root six multiplies with itself to give four times six. And the two cross terms cancel, as desired.

The denominator has been rationalized, that is to say, converted from an irrational to a rational number. We now have 104 root 19 plus 104 root six all over 52. Notice that this 52 is a common factor in the numerator and the denominator.

After this final simplification, let’s clear away our workings to make space. Since π‘₯ is equal to two root 19 plus two root six and 𝑦 is its conjugate, we can see that 𝑦 equals two root 19 minus two root six.

We can now proceed with the calculation. Take a moment to notice though that if we first square π‘₯ and then square 𝑦 and then multiply the results, we’re going to end up with lots of horrible square roots all over the place. A much better plan is to first use the law of indices, which tells us that π‘₯ squared 𝑦 squared is the same as π‘₯𝑦 squared. Because 𝑦 is conjugate to π‘₯, we know that π‘₯𝑦 will be a rational number. In fact, we’ve already done this calculation; it’s 52.

Our final answer is therefore 52 squared, which is 2704.

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