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# If n is an integer greater than 1, what is the value of 10(root10)

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Math Expert
Joined: 02 Sep 2009
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If n is an integer greater than 1, what is the value of 10(root10)  [#permalink]

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17 Jun 2020, 04:48
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If n is an integer greater than 1, what is the value of $$10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{...}}}}$$, where the given expression extends to an infinite number of roots?

A. 10

B. $$10^{\frac{1}{n}}$$

C. $$10^{\frac{n-1}{n}}$$

D. $$10^{\frac{n}{n-1}}$$

E. $$10^{n}$$

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Re: If n is an integer greater than 1, what is the value of 10(root10)  [#permalink]

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17 Jun 2020, 04:50
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Re: If n is an integer greater than 1, what is the value of 10(root10)  [#permalink]

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17 Jun 2020, 08:35
Bunuel wrote:

If you look at all of those questions, you'll notice not one of them is official. The GMAT never tests infinite sums and products like the ones you see in these questions, because to answer questions like this, first you need to establish that these infinite expressions "converge" (they don't become infinite or undefined) and to do that, you need calculus. Calculus is not tested on the GMAT.

So these questions are all out of scope, and I don't see how the technique one uses to solve them could be relevant to any actual GMAT question, but if we need to solve, we can do as follows. Let x equal the expression:

$$x = 10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{...}}}}$$

Now notice that after the first "$$\sqrt[n]{ }$$", what we see is identical to x. So we can replace it with "x", and we have

$$x = 10*\sqrt[n]{x}$$

Now we just need to solve this equation: x/10 = x^(1/n), so x^n / 10^n = x, and x^(n-1) = 10^n, so x = 10^(n/(n-1))

With these answer choices, it's even faster to notice that we're multiplying 10 here by something greater than 1. So the answer must be greater than 10, and so must be D or E. But the larger n is, the smaller the roots all get, so the smaller the eventual answer should be. So E can't be right, because E gets bigger if you make n bigger, and only D is possible.
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Re: If n is an integer greater than 1, what is the value of 10(root10)  [#permalink]

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17 Jun 2020, 09:04
1
Bunuel wrote:
If n is an integer greater than 1, what is the value of $$10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{...}}}}$$, where the given expression extends to an infinite number of roots?

A. 10

B. $$10^{\frac{1}{n}}$$

C. $$10^{\frac{n-1}{n}}$$

D. $$10^{\frac{n}{n-1}}$$

E. $$10^{n}$$

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Asked: If n is an integer greater than 1, what is the value of $$10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{...}}}}$$, where the given expression extends to an infinite number of roots?

Let x = $$10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{...}}}}$$

$$x = 10*\sqrt[n]{x}$$
$$x^n = 10^n * x$$
$$x^n - 10^n * x = 0$$
$$x^{n-1} - 10^n = 0$$ since $$x \neq 0$$
$$x^{n-1} = 10^n$$
$$x = 10^{\frac{n}{n-1}}$$

IMO D
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Re: If n is an integer greater than 1, what is the value of 10(root10)  [#permalink]

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17 Jun 2020, 23:19
Bunuel wrote:
If n is an integer greater than 1, what is the value of $$10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{...}}}}$$, where the given expression extends to an infinite number of roots?

A. 10

B. $$10^{\frac{1}{n}}$$

C. $$10^{\frac{n-1}{n}}$$

D. $$10^{\frac{n}{n-1}}$$

E. $$10^{n}$$

Solution

• Given n is an integer greater than 1
o This means, $$\frac{1}{n} < 1$$
• Now, $$10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{….,}}}} = 10*10^{\frac{1}{n}}*10^{\frac{1}{n}*\frac{1}{n}}*10^{\frac{1}{n}*\frac{1}{n}*\frac{1}{n}}* ……,$$
o $$⟹ 10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{….,}}}} = 10^{1+\frac{1}{n} + \frac{1}{n^2} +\frac{1}{n^3}…..}$$
• We can observe that $${\frac{1}{n} + \frac{1}{n^2} +\frac{1}{n^3}…..}$$ is an infinite G.P.,
o $$⟹ 10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{….,}}}} = 10^{1+\frac{1}{n} + \frac{1}{n^2} +\frac{1}{n^3}…..}$$ = $$10^{\frac{1}{(1-\frac{1}{n})}}$$ = $$10^{\frac{n}{n-1}}$$

Thus, the correct answer is Option D.
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Re: If n is an integer greater than 1, what is the value of 10(root10)  [#permalink]

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18 Jun 2020, 02:38
n= 1
10*sqrt(10(sqrt10sqrt10..
10*3.32sqrt (3.32)* xxx= results seems infinity

A. 10
B. 10
C. 1
D. 1/10)^1-1/n= infinity
E. 10

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Re: If n is an integer greater than 1, what is the value of 10(root10)  [#permalink]

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20 Jun 2020, 14:30
Bunuel wrote:
If n is an integer greater than 1, what is the value of $$10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{...}}}}$$, where the given expression extends to an infinite number of roots?

A. 10

B. $$10^{\frac{1}{n}}$$

C. $$10^{\frac{n-1}{n}}$$

D. $$10^{\frac{n}{n-1}}$$

E. $$10^{n}$$

Solution:

Letting x = the expression, we can create the equation:

x = 10 * n^√x

x = 10x^(1/n)

x^(1 - 1/n) = 10

x^[(n - 1)/n] = 10

x = 10^[n/(n - 1)]

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Re: If n is an integer greater than 1, what is the value of 10(root10)   [#permalink] 20 Jun 2020, 14:30

# If n is an integer greater than 1, what is the value of 10(root10)

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