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If π§ one is equal to negative one over root two minus π and π§ two is equal to negative one-half plus root two π, is it true that π§ one squared is equal to π§ two?
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Here, weβve been given two complex numbers represented in algebraic or rectangular form.
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And weβre being asked to decide whether the statement π§ one squared is equal to π§ two is true.
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To work this out, weβre going to need to first evaluate π§ one squared.
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Thatβs negative one over root two minus π all squared.
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We recall that squaring a number is the same as multiplying it by itself.
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So π§ one squared is equal to negative one over root two minus π multiplied by negative one over root two minus π.
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And once we donβt want to consider π as a variable, we can distribute these brackets as normal.
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Letβs look at the FOIL method.
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F stands for first.
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We multiply the first term in the first bracket by the first term in the second bracket.
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One multiplied by one is one.
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And root two multiplied by root two is two.
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And since weβre multiplying a negative by a negative, we end up with positive one-half.
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We then multiply the outer term in each bracket.
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Thatβs negative one over root two multiplied by negative π.
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That gives us positive π over root two.
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We repeat this process with the inner terms, which once again gives us π over root two.
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And L stands for last.
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We multiply the last term in each bracket.
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Thatβs π squared.
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Now remember, we said π is not a variable.
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Itβs in fact the square root of negative one.
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This means that π squared is actually equal to negative one.
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So our expression for π§ one squared becomes one-half plus two lots of π over root two minus one.
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Now, one-half minus one is negative one-half.
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So currently, we can see that π§ one squared is equal to negative one-half plus two π over root two.
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And we need to rationalize the denominator of this second fraction.
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To do this, we multiply both the numerator and the denominator by the square root of two.
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Two π multiplied by root two is two root two π.
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And the square root of two multiplied by the square root of two is two.
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So this second fraction becomes two root two π over two.
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But of course, we can simplify this further by dividing through by two.
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So we see that π§ one squared is equal to negative one-half plus root two π.
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And we said that π§ two was negative one-half plus root two π.
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So we can see that π§ one squared is equal to π§ two.
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And this statement is indeed true.