Video: Evaluating Algebraic Expressions Using the Properties of Conjugate Irrational Numbers

Given that π‘₯ = 5/(√7 βˆ’ √2) and 𝑦 = √7 βˆ’ √2, find π‘₯ + 𝑦 expressing your answer in simplest form.

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Video Transcript

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

Before we can add π‘₯ and 𝑦, we need to rationalize the fraction π‘₯ β€” five divided by root seven minus root two. In order to rationalize this, we must multiply the top β€” the numerator β€” and the bottom β€” the denominator β€” by root seven plus root two. This gives us five multiplied by root seven plus root two divided by root seven minus root two multiplied by root seven plus root two.

Expanding the parenthesis on the numerator gives us five root seven plus five root two. In order to expand the parentheses on the bottom, we need to use the FOIL method. Multiplying the first terms root seven multiplied by root seven gives us seven. Multiplying the outside terms root seven multiplied by root two gives us root 14. Multiplying the inside terms gives us negative root 14. And finally, multiplying the last terms gives us negative two.

As the root 14s cancel, we’re left with seven minus two, which is equal to five. Therefore, the denominator is equal to five. All three of these terms are divisible by five. This means that π‘₯ can be rewritten as root seven plus root two as five root seven divided by five is root seven and five root two divided by five is root two.

We were told in the question that 𝑦 was equal to root seven minus root two. We now need to add these two expressions. Adding the two expressions gives us root seven plus root two plus root seven minus root two. Root seven plus root seven is equal to two root seven and root two minus root two is equal to zero.

This means that the expression for π‘₯ plus 𝑦 in its simplest form is two root seven.

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