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Can anyone please explain to me how this happens!? My brain cannot think how to get from one to the other. Thanks!

$8 {\sqrt 2}$ to $ \frac{16} {\sqrt 2}$

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4 Answers 4

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$$8\sqrt{2}=8\sqrt{2}\times 1=8\sqrt{2}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{8\sqrt{2}\times\sqrt{2}}{\sqrt{2}}=\frac{8\times2}{\sqrt{2}}=\frac{16}{\sqrt{2}}$$

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  • $\begingroup$ Thank you!!! So simply I completely overlooked it. $\endgroup$
    – Dani
    Aug 2, 2014 at 7:17
  • $\begingroup$ @Dani Glad to help $\endgroup$ Aug 2, 2014 at 7:24
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This happens often to so many students of mine. Always remember this:

$$\sqrt{2} = \frac{2}{\sqrt{2}}$$

In this case, it means $8 \cdot \sqrt{2} = 8 \cdot \frac{2}{\sqrt{2}} = \frac{16}{\sqrt{2}}$

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Hint: multiply the "top" and "bottom" by $\sqrt{2}$

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The following is not a direct answer but a personal way of thinking about irrational numbers a bit more naturally:

I think people are slightly afraid of irrational numbers because it's difficult to imagine them. If you were to search for $\sqrt 2$ on a number line, you'd be looking in between $1$ and $2$, where you'll be looking between $1.4$ and $1.5$, where you'll be looking in between $1.41$ and $1.42$, then between $1.414$ and $1.415$ and so on until the number you've found when multiplied with itself gives $2$. (which is something that no one who is rational would try to do. This is one of the reasons, I think, that such numbers are called irrational)

It's confusing to imagine the existence of such a number with an infinite amount of digits. But it does exist and it's nothing to be afraid of.

In your question, you wished to convert $8\sqrt 2$ to $\frac{16}{\sqrt 2}$

If you find it difficult to imagine their equality, then look at how I take advantage of the fact that for every real number $x$, $ \sqrt{x^2} = |x|$:

$$ 8\sqrt 2 = \sqrt{(8\sqrt 2)^2} = \sqrt{64\times 2} = \sqrt {128}$$ Similarly, $$ \frac{16}{\sqrt 2} = \sqrt{({\frac{16}{2}})^2} = \sqrt{\frac{256}{2}} = \sqrt{128}$$

From the above, it is hopefully clear that $$ 8\sqrt 2 = \sqrt{64 \times 2} = \sqrt{64 \times \frac{4}{2}}= \sqrt{\frac{256}{2}} = \frac{16}{\sqrt 2 }$$

The middle part that connects the two takes use of the idea that a ratio can be scaled up or down by any multiple. Thus, a ratio $2:1$ can be scaled up by $2$ as $4:2$

This is the same property that the other answers have used,
$8\sqrt 2 : 1$ had been scaled up by $\sqrt 2$ to $16 : \sqrt 2$

Also, if you familiarize yourself with the powers of $2$, you would realize that $2^3 = 8$ and $2^4 = 16$ and by the laws of exponents, you can easily come to the same conclusion.

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