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What is the right method for solving a problem like this:

”$4\sqrt{5}$ is the same as which square root?"

Possible answers are:

  • $\sqrt{20}$
  • $\sqrt{10}$
  • $\sqrt{40}$
  • $\sqrt{80}$

I have been informed that $\sqrt{80}$ is the right answer, but I do not understand why.

Asaf Karagila
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Big Swede
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    I now noticed that GeorgSalibe kindly adjusted the format of my question to how it should be used on this site. May I ask for a link or some other information about this procedure, so maybe I can do it correct from the beginning myself next time! – Big Swede Jul 26 '15 at 14:09
  • Hi! You would have to learn LaTeX, it's very easy. You can learn the basic mathematical operations in a day. – Naz Jul 26 '15 at 15:21
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    See http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – PM 2Ring Jul 26 '15 at 15:54
  • About your other question, I simply could not understand why you cannot just use paper and pencil/pen. It is not that the multiplication table needs any computing power, since you know all the possible combinations and there are so few of them. I never intended any rudeness but I do intend that you seriously consider doing things without a computing device whenever reasonable. In fact, by the time you prepare a multiplication table on paper you probably would have memorized large chunks of it. Often the best way to get familiar with something is to do it all by yourself. – user21820 Aug 12 '15 at 09:41
  • Generally we are in an agreement about this. And I notice that you never ment any rudness, so this is now OK for me. There are certain reasons why I, in this case, would like to use computer power. That does not matter now, but if you want to know the exact reason you can send me a personal message and I will explain. – Big Swede Aug 12 '15 at 11:23

6 Answers6

8

Because $4=\sqrt{16}$. Then $\sqrt{16}\cdot \sqrt{5} = \cdots$

peterwhy
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    OK! So the point was to understand/remember that 4=√16, and then the head arithmetics will be something like: 1) 105=50 2) 65= 30, 3) 50+30=80 4) Put the result under the radical sign = √80. And thats it? OK! Thank you! – Big Swede Jul 26 '15 at 13:48
  • @BigSwede Yes. Until you have negative number under radical sign... – peterwhy Jul 26 '15 at 14:06
  • I see! I will come back if I must solve that kind of problem... =) – Big Swede Jul 26 '15 at 14:12
5

I prefer this formulation, particularly for beginners:

$x=4 \times \sqrt{5} = \sqrt{16} \times \sqrt{5} = \sqrt{16\times5} = \sqrt{80}$

This is how I did it in my head when reading your question.

It's also good practice for solving proofs when you have to appear to go in the wrong direction for a moment (making $4$ lexically "larger", as $\sqrt{16}$) in order to bring terms together.

Lightness Races in Orbit
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  • I like the formulation: "you have to appear to go in the wrong direction". Usually, the question is to simplify an expression and students are not aware that equality is a reflexive relation. So they are surprised when in some proofs we add and subtract the same quantity, whether it be $1 - 1$ or $-f(x+h)f(x)+f(x+h)f(x)$. – MasB Jul 26 '15 at 18:14
  • @BernardMassé: Yep, exactly. That was what I was (badly) trying to say. :) I was always quite good at those approaches in my youth, and for some reason found them very satisfying. – Lightness Races in Orbit Jul 26 '15 at 18:15
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$4\sqrt{5} = \sqrt{x} \Leftrightarrow \sqrt{4^25} = \sqrt{x}$

since $\sqrt{x}$ is injective $x = 4^25 = 80$

zzchan
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    Thank you! This is maybe embarassing, but I do not understand the symbol "⇔". I do not understand the meaning of "injective" in this context either. Can you please explain? – Big Swede Jul 26 '15 at 13:58
  • $\Leftrightarrow$ means equivalent. An injective function takes different input values to different output values so if output is equal the input must also be equal. – zzchan Jul 26 '15 at 14:30
  • OK, fine! I understand the part about "⇔". (I only partially understand the part about injective functions, but that does not matter at this Point!)So, thank you, we can stop there! – Big Swede Jul 26 '15 at 15:10
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Hint:

use: $a\sqrt{b}=\dfrac{a}{|a|} \sqrt {a^2b}= \mbox{sign}(a)\sqrt {a^2b}$

Here $|a|$ is the absolute value of $a$, so that $\dfrac{a}{|a|}= 1$ if $a>0$ and $\dfrac{a}{|a|}= -1$ if $a<0$. This same result can be represented wit the function "sign (a)" that is simply the sign of $a$.

The use of this function is very important when we work with radicals because the square of a number is always positive, so, squaring $a$ we can forget that the basis of the square was negative.

In you case $a=4$, so sign$(4)=+1$ and you have no problem writing: $$ 4 \sqrt{5}=\sqrt{16 \times 5} $$

but if you have , e.g. : $a=-4$ the correct result is:

$$ -4 \sqrt{5}=-\sqrt{16 \times 5} $$

Emilio Novati
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$x=4\sqrt 5\implies x^2=16\cdot 5=80\implies x=\sqrt{80}$

ajotatxe
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  • Thank you! I actually understand the answer, but not the meaning of, or general usage of, the symbols "x", "⟹" and "x 2" in this context. Can you please explain? – Big Swede Jul 26 '15 at 14:03
  • +1: This is the best approach, but (unfortunately, the most opaquely put. @BigSwede: The idea is to show that $$\left(4\sqrt5\right)^2=4^2\sqrt5^2=16\cdot5=80.$$ Hence, since $4\sqrt5$ is positive (why?), and since $4\sqrt5$ is a square root of $80$ *by definition*, then $4\sqrt5=\sqrt{80}.$ – Cameron Buie Jul 26 '15 at 18:21
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Notice, in general, for any two positive real numbers $a$ & $b$ $$a\sqrt{b}=\sqrt{a^2b}$$ Hence, we have $$4\sqrt{5}=\sqrt{(4)^2(5)}$$ $$=\sqrt{16\times 5}=\sqrt{80}$$ Yes $\sqrt{80}$ is right answer.

Harish Chandra Rajpoot
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