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Given that π₯ is equal to root seven divided by root 13 and π¦ is equal to root 13 divided by root seven, find 182 multiplied by π₯ plus π¦, expressing your answer in simplest form.
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Our first step here is to rationalize the denominator of both π₯ and π¦.
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In order to rationalize the denominator of π₯, we multiply the top and bottom of the fraction by root 13.
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At this point, it is worth remembering two of the laws of surds.
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Root π multiplied by root π is equal to π, and root π multiplied by root π is equal to root ππ.
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In our question, root seven multiplied by root 13 will be equal to root 91, as seven multiplied by 13 equals 91.
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Multiplying the denominators gives us 13, as root 13 multiplied by root 13 is equal to 13.
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π¦ was equal to root 13 divided by root seven.
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To rationalize this surd, we multiply the top and bottom by root seven.
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Once again, root 13 multiplied by root seven is equal to root 91, and root seven multiplied by root seven is equal to seven.
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We now have simplified expressions for π₯ and π¦.
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We want to calculate 182 multiplied by π₯ plus π¦.
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Letβs first work out what π₯ plus π¦ would be.
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π₯ plus π¦ is equal to root 91 divided by 13 plus root 91 divided by seven.
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In order to add two fractions, we need to find a common denominator, in this case a number that is in the seven and 13 times table.
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The lowest common denominator in this case is 91.
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13 multiplied by seven is equal to 91.
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Multiplying the numerator and denominator of the first fraction by seven gives a seven root 91 divided by 91.
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Multiplying the numerator and denominator of the second fraction by 13 gives us 13 root 91 divided by 91.
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Now that the denominators are the same, we can add the two numerators.
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Seven root 91 plus 13 root 91 is equal to 20 root 91.
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Therefore, π₯ plus π¦ is equal to 20 root 91 divided by 91.
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We now need to multiply this expression by 182 to get the final answer in this question.
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182 divided by 91 is equal to two, as 91 is a half of 182.
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Weβre therefore left with two multiplied by 20 root 91.
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This is equal to 40 root 91.
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If π₯ is equal to root seven divided by root 13 and π¦ is equal to root 13 divided by root seven, then 182 multiplied by π₯ plus π¦ is equal to 40 root 91.