If n is an integer greater than 1, what is the value of 10* [n]10* [n]10* [n]10* [n]..., where the given expression extends to an infinite number of roots?

If n is an integer greater than 1, what is th… | GMAT Sprint

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PS · Problem Solving

705-805

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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?

Correct answer marked below

10^{\frac{1}{n}}

10^{\frac{n-1}{n}}

10^{\frac{n}{n-1}}

Correct

10^{n}

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Explanation

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Let

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

.

Since the expression is infinite, we can write:

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

.

Raise both sides to the power of n:

x^n = 10^n * x

.

Divide both sides by x (x > 0):

x^{n-1} = 10^n

.…

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What this question tests

Your handling of square and higher roots, including simplifying radicals.

Forgetting the ± when taking a square root to solve an equation, or assuming √(a²) = a.