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Given that π§ sub one is equal to two root three plus two π and π§ sub two is equal to negative two minus two root three π, find π§ sub one multiplied by π§ sub two, giving your answer in exponential form.
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In this question, weβre given two complex numbers in the form π₯ plus π¦π, where π₯ is the real part and π¦ is the imaginary part of the complex number.
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Weβre asked to calculate the product of the two complex numbers π§ sub one and π§ sub two.
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We can do this by distributing the parentheses using the FOIL method.
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Multiplying the first terms gives us negative four root three.
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Multiplying the outer or outside terms gives us negative 12π.
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This is because two multiplied by negative two is negative four.
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Root three multiplied by root three is equal to three.
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Finally, negative four multiplied by three is equal to negative 12.
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Multiplying the inner or inside terms gives us negative four π.
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Finally, multiplying the last terms gives us negative four root three π squared.
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We recall that when dealing with complex numbers, π squared is equal to negative one.
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This means that negative four root three π squared is equal to positive four root three.
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We can now collect like terms.
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The real terms cancel as negative four root three plus four root three is equal to zero.
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π§ one multiplied by π§ two is therefore equal to negative 16π.
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We recall that any complex number can be written in exponential form such that π§ is equal to π multiplied by π to the ππ, where π is the modulus of the complex number and π is its argument.
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The modulus of a complex number is equal to the square root of π₯ squared plus π¦ squared, where π₯ and π¦ are the real and imaginary parts, respectively.
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The argument π is equal to the inverse tan of π¦ over π₯.
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As there is no real part to the complex number negative 16π, then π₯ is equal to zero, and π¦ is equal to negative 16.
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π is therefore equal to the square root of zero squared plus negative 16 squared.
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This is equal to 16.
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π is equal to the inverse tan of negative 16 over zero.
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As the denominator is equal to zero, this will be undefined.
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We are looking for the value of π where tan π is undefined.
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This occurs at π over two plus ππ.
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In order to work out the correct value of π for this question, we will clear some space and draw the Argand diagram.
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The complex number π§ one π§ two had a real value equal to zero and an imaginary value equal to negative 16.
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This lies at the point which separates the third and fourth quadrants.
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We know that this corresponds to the angle three π over two.
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The value of the argument π is three π over two.
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We can therefore conclude that the complex number π§ one π§ two written in exponential form is equal to 16π to the three π over two π.