WEBVTT
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Given that π₯ is equal to five divided by root seven minus root two and π¦ is equal to root seven minus root two, find π₯ plus π¦ expressing your answer in simplest form.
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Before we can add π₯ and π¦, we need to rationalize the fraction π₯ β five divided by root seven minus root two.
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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.
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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.
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Expanding the parenthesis on the numerator gives us five root seven plus five root two.
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In order to expand the parentheses on the bottom, we need to use the FOIL method.
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Multiplying the first terms root seven multiplied by root seven gives us seven.
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Multiplying the outside terms root seven multiplied by root two gives us root 14.
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Multiplying the inside terms gives us negative root 14.
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And finally, multiplying the last terms gives us negative two.
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As the root 14s cancel, weβre left with seven minus two, which is equal to five.
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Therefore, the denominator is equal to five.
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All three of these terms are divisible by five.
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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.
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We were told in the question that π¦ was equal to root seven minus root two.
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We now need to add these two expressions.
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Adding the two expressions gives us root seven plus root two plus root seven minus root two.
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Root seven plus root seven is equal to two root seven and root two minus root two is equal to zero.
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This means that the expression for π₯ plus π¦ in its simplest form is two root seven.