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I'm reading through some texts and came across the following two lines. I'm curious how they should be read.

Volume = $4/3\enspace π r^3$

Surface Area = $4\enspace π r^2$

In the example of Volume: Is $4$ divided by $3$, multiplied by $\pi$? Is it $4/3$ of $\pi$? What's the suggested way of reading stuff like this, because it comes off as poorly-structured.

Kumar
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David Metcalfe
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  • The two ways you suggest give the same value. – Improve Aug 05 '19 at 18:58
  • volume = (4/3) π r^3 – ashishyadaveee11 Aug 05 '19 at 18:59
  • Possibly related: 8 / 4 (4-2) = ?. It appears you might be confused as to whether the $\pi r^3$ should be on the denominator or on the numerator. In this case specifically, it is $\frac{4}{3} \times \pi \times r^3$, not $\frac{4}{3\pi r^3}$ or similar. Also note that the cube applies only to $r$ here, it is $\frac{4}{3}\times\pi\times (r^3)$, not something like $(\frac{4}{3}\times \pi\times r)^3$ – JMoravitz Aug 05 '19 at 19:03
  • Side note written as is hints at $\frac{dV}{dr}=S$. It's always useful to think why something is written the way it is if there are multiple possibilities. – Karl Aug 05 '19 at 20:25

2 Answers2

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It's $$V=\frac{4}{3}\pi r^3=\frac{4}{3}\cdot\frac{\pi r^3}{1}=\frac{4\cdot\pi r^3}{3\cdot1}=\frac{4(\pi r^3)}{3}=\frac{(4\pi) r^3}{3}=\frac{4\pi r^3}{3}.$$

Michael Rozenberg
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  • Can you provide a simpler breakdown to go with this? @Jamminermit's answer is more my speed. – David Metcalfe Aug 05 '19 at 19:02
  • @David Metcalf I wrote number of equalities. Which of them is not clear? – Michael Rozenberg Aug 05 '19 at 19:06
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    @DavidMetcalfe what specifically is unclear about this? This answer is effectively pointing out that $4\pi r^3$ could be read and thought of as $4$ times $\pi r^3$ or it could be thought of as $4\pi$ times $r^3$ or it could be thought of as $4$ times $\pi$ times $r^3$, etc... due to the associative nature of multiplication, so pick the one that appeals most to you at the time to use. – JMoravitz Aug 05 '19 at 19:08
  • @MichaelRozenberg What you provided is nifty, and probably fully answers the question to those savvy in their math, but it gets no closer to an answer for a layperson. I don't know how to read what you wrote. So, I'm asking if you can write it as though you were speaking it, akin to what Jamminermit did. Minimal use of symbols. – David Metcalfe Aug 05 '19 at 19:43
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It’s just the constant pi multiplied by 4/3, multiplied by the radius cubed. Since multiplication is commutative, you can do it in any order.

Jamminermit
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  • Explain please, how you used a commutativity? If you'll see down-voting it's not mine. – Michael Rozenberg Aug 05 '19 at 19:04
  • Commutativity basically applies to operations such as addition and (in this case) multiplication. It basically just means that a+b=b+a (the same is true with multiplication). So in summary I have just stated commutativity as the order you carry out the calculations does not matter. – Jamminermit Aug 05 '19 at 20:06