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Find the multiplicative inverse of the square root of six over 30.
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Remember, the multiplicative inverse of a number 𝑎 is the number that when we multiply it by 𝑎 gives us one.
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Another way of considering this is as the reciprocal of that number.
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So, if we have a number 𝑎, its reciprocal is one over 𝑎.
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So, here, we need to find the number that when we multiply it by the square root of six over 30, we get one.
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If we let 𝑥 be the multiplicative inverse of the square root of six over 30, then we could say that 𝑥 times the square root of six over 30 is equal to one.
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Now, equivalently, we would achieve this by solving this equation.
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We said that it is also the reciprocal of the original number.
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That’s one over the number.
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So, it’s one over the square root of six over 30.
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This doesn’t look very nice, though.
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So, we’re going to recall how we divide fractions.
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Really, we’re wanting to divide one by the square root of six over 30.
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So, we write one as one over one.
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And then recall that to divide by a fraction, we multiply by the reciprocal of that fraction.
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This is sometimes called keep, change, flip.
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So, 𝑥 is equal to one over one times 30 over root six.
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And if we multiply the numerators and then separately multiply the denominators of our fractions, we get 𝑥 is equal to 30 over root six.
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Now, in fact, we really didn’t need to perform this step.
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Given a fraction in the form 𝑎 over 𝑏, its reciprocal is simply 𝑏 over 𝑎.
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But of course, it’s always good to understand where these things come from.
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So, we found the multiplicative inverse to be 30 over the square root of six.
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But we’re really not finished.
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We need to rationalize the denominator.
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In other words, we want the denominator of our fraction to be rational.
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At the moment, it’s an irrational number.
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The square root of six cannot be written as a fraction where the numerator and denominator are integers.
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So, how do we achieve this?
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Well, we multiply the numerator and denominator of our fraction by the square root of six.
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That’s the same as multiplying by the square root of six over the square root of six, or just by multiplying by one.
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And in doing so, all we’re doing is creating an equivalent fraction.
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30 times the square root of six is 30 root six.
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Then, the square root of six times itself is, of course, simply six.
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Multiplying a number by itself is squaring it and squaring is the inverse to square rooting.
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So, we see that our multiplicative inverse is 30 root six over six.
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Finally, we spot that both 30 and six have a common factor of six.
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And so, dividing through by six, we get five root six over one, which is simply five root six.
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The multiplicative inverse of the square root of six over 30 is five root six.