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# Find the value of x

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VP
Joined: 18 May 2008
Posts: 1107
Find the value of x  [#permalink]

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04 Feb 2009, 01:04
3
18
00:00

Difficulty:

35% (medium)

Question Stats:

70% (01:46) correct 30% (02:09) wrong based on 759 sessions

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Find the value of x

$$x= \sqrt[2]{20+\sqrt[2]{20+\sqrt[2]{20}}}$$

1. 20
2. 5
3. 2
4. 8
Math Expert
Joined: 02 Sep 2009
Posts: 51280
Find the value of x  [#permalink]

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29 Sep 2010, 06:45
7
8
nonameee wrote:
Bunuel, would you be so kind and look at this question. Is there any other way to solve it rather than elimination? Can you describe elimination in greater detail? Thank you.

Find the value of x

$$x= \sqrt{20+\sqrt{20+\sqrt{20}}}$$

1. 20
2. 5
3. 2
4. 8

Question should be what is the approximate value of $$x$$.

Obviously answer choice C (2) is out as $$\sqrt{20+some \ #}>4$$.

Now, $$4<\sqrt{20}<5$$:
$$x= \sqrt{20+\sqrt{20+\sqrt{20}}}= \sqrt{20+\sqrt{20+(# \ less \ than \ 5)}}= \sqrt{20+\sqrt{# \ less \ than \ 25}}= \sqrt{20+(# \ less \ than \ 5)}=$$

$$=\sqrt{# \ less \ than \ 25}=# \ less \ than \ 5\approx{5}$$.

Next, exactly 5 to be the correct answer question should be:

If the expression $$x=\sqrt{20+{\sqrt{20+\sqrt{20+\sqrt{20+...}}}}}$$ extends to an infinite number of roots and converges to a positive number x, what is x?

$$x=\sqrt{20+{\sqrt{20+\sqrt{20+\sqrt{20+...}}}}}$$ --> $$x=\sqrt{20+({\sqrt{20+\sqrt{20+\sqrt{20+...})}}}}$$, as the expression under square root extends infinitely, then expression in brackets would equal to $$x$$ itself so we can rewrite given expression as $$x=\sqrt{20+x}$$. Square both sides $$x^2=20+x$$ --> $$x=5$$ or $$x=-4$$. As given that $$x>0$$ then only one solution is valid: $$x=5$$.

Hope it helps.
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Joined: 15 Jan 2009
Posts: 45
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04 Feb 2009, 04:41
1
1
ritula wrote:
Find the value of x

$$x= \sqrt[2]{20+\sqrt[2]{20+\sqrt[2]{20}}}$$

1. 20
2. 5
3. 2
4. 8

Since the answer choices are dissimilar, we can estimate the answer choice here. The $$\sqrt[2]{20}$$ is somewhere between 4 and 5. Suppose it's 5, then we'll get
$$\sqrt[2]{20+\sqrt[2]{20+5}$$=$$\sqrt[2]{20+5}$$=$$5$$
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04 Feb 2009, 07:29
agree with 5..

i estimated it to be 5..

now if they had a 4 in the ans choices..that would have been tough..
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04 Feb 2009, 07:32
FN wrote:
agree with 5..

i estimated it to be 5..

now if they had a 4 in the ans choices..that would have been tough..

even if you have 4.. its not tough.. sqrt(20) clearly.. >4

I agree if answer choice has options like 4.9 or 4.8...
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04 Feb 2009, 07:38
ritula wrote:
How did u get 5?
quote="scthakur"]B. 5 (by process of elimination).
[/quote]

actual value of x= 4.994690378 ~5

only way to do is process of elimination.
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04 Feb 2009, 07:59
1
3
ritula wrote:
Find the value of x

$$x= \sqrt[2]{20+\sqrt[2]{20+\sqrt[2]{20}}}$$

1. 20
2. 5
3. 2
4. 8

1: $$x= \sqrt{20+\sqrt{20+\sqrt{20}}}$$
$$x= \sqrt{20+\sqrt{20+4.47}}$$
$$x= \sqrt{20+\sqrt{24.47}}$$
$$x= \sqrt{20+4.95}$$
$$x= \sqrt{24.95}$$
$$x= 4.995 = approx. 5.00$$

2: From third sqrt: sqrt 20 = 4 and a fraction of 1.
From second sqrt: sqrt (20+4.00 = 24) = 4 and a fraction of 1.
From first sqrt: sqrt (20+4.00 = 24) = 4 and a fraction of 1. which is definitely close to none other than 5.

3: Using POE:

A: it cannot be 20 cuz for 20, the value under root must be 400, which is impossible. so A is ruled out.
B: It could be 5 as done above in method 2.
3. 2 is also not possible because even if we consider first 20 under root, the value must not be smaller than 4.
4. 8 is not possible because for 8, the value under root must be 64. Even if we add up all three 20s, the sum would not be more than 60. so it is also not possible. So left with 5.

So B make sense.
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04 Feb 2009, 11:39
1
If so, shouldn't the question be "What is the approximate value of x?" Just curious
Intern
Joined: 27 Sep 2009
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02 Oct 2009, 08:49
x=root of 20+x
x^2=20+x
x^2-x=20
solving we get x=5,-4
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02 Oct 2009, 11:53
boiled it down to roughly 5.5ish. 8 is too high. must be 5.
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29 Sep 2010, 05:15
Bunuel, would you be so kind and look at this question. Is there any other way to solve it rather than elimination? Can you describe elimination in greater detail? Thank you.
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04 Oct 2010, 02:31
Bunuel, thank you very much.
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Re: Find the value of x  [#permalink]

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22 Sep 2013, 03:02
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Re: Find the value of x  [#permalink]

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11 Dec 2014, 01:37
$$x= \sqrt[2]{20+\sqrt[2]{20+\sqrt[2]{20}}}$$

Converting the square root sign to power

$$x = (20 + (20 + (20)^{\frac{1}{2}})^{\frac{1}{2}})^{\frac{1}{2}}$$

Squaring both sides

$$x^2 = 20 + (20 + (20)^{\frac{1}{2}})^{\frac{1}{2}}$$

Now start looking at the OA

Value of $$x^2$$ has to be around 25, but way less than 64

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Re: Find the value of x  [#permalink]

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08 Jul 2016, 14:49
why is it that only 4 options given in the question?
I think that this is not a gmat type question
as for explanation,I agree with bunuel that it needs to be asking approximate value of x
Correct Ans - B
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Re: Find the value of x  [#permalink]

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08 Jul 2016, 21:06
Fortunately there is a super short cut to solve this question (that shortcut is applicable to this question atleast )
This whole expression can be seen as
$$x= \sqrt[2]{20+\sqrt[2]{some number}$$

.
Square root of 20 is 4.4 approx and the rest of the nested square root will be a small decimal value because they are square root inside square root

so the answer will be 4.4 + 0.some value

Look out for a nearest answer greater than 4.4

It's 5 in the given question

If you get this kind of questions in GMAT , thank your stars because you can save tremendous amount of time and add sure marks to your score.

ritula wrote:
Find the value of x

$$x= \sqrt[2]{20+\sqrt[2]{20+\sqrt[2]{20}}}$$

1. 20
2. 5
3. 2
4. 8

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Re: Find the value of x  [#permalink]

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01 Jul 2018, 07:38
ritula wrote:
Find the value of x

$$x= \sqrt[2]{20+\sqrt[2]{20+\sqrt[2]{20}}}$$

1. 20
2. 5
3. 2
4. 8

Am I wrong to convert this to $$20^\frac{1}{2}$$ + $$20^\frac{1}{4}$$ + $$20^\frac{1}{8}$$ ?
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Re: Find the value of x &nbs [#permalink] 01 Jul 2018, 07:38
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