How many integer values of n satisfy the ineq… | GMAT Sprint

Free Practice Question

PS · Problem Solving

705-805

Roots

How many integer values of n satisfy the inequality $3\sqrt{n} > \sqrt{n^3}$ ?

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Correct answer marked below

Correct

Infinitely many

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Explanation

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Official Explanation

Official

We start with the inequality:

3\sqrt{n} > \sqrt{n^3}

.

First, note that

\sqrt{n^3}

can be written as

\sqrt{n \cdot n \cdot n} = n\sqrt{n}

.

Substituting this into the inequality gives:

3\sqrt{n} > n\sqrt{n}

.

Rearranging terms:

3\sqrt{n} - n\sqrt{n} > 0

.

Factor out

\sqrt{n}

\sqrt{n}(3 - n) > 0

.

Since

\sqrt{n}

cannot be negative (and from the inequality, n cannot be 0, otherwise both sides would be 0, leading to 0 > 0), we have

\sqrt{n} > 0

. Thus, the inequality reduces to

3 - n > 0

, which gives

n < 3

.…

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What this question tests

What it tests

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

Common trap

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

How many integer values of n satisfy the inequality 3n>n33\sqrt{n} > \sqrt{n^3}3n​>n3​?