What is the value of 3 5 3 5 ..., where the given expression extends to an infinite number of roots?

What is the value of 3 5 3 5 ..., where the g… | GMAT Sprint

Free Practice Question

PS · Problem Solving

705-805

Roots

What is the value of $3\sqrt{5\sqrt{3\sqrt{5\sqrt{...}}}}$ , where the given expression extends to an infinite number of roots?

Correct answer marked below

3\sqrt{5}

3\sqrt[3]{75}

Correct

3\sqrt{75}

You can see the correct answer above. Sign in for free to unlock the complete worked solution.

Get detailed analytics, unlock full explanations, and move up difficulty tiers as you practice.

Explanation

Preview

Official

Let

x = 3\sqrt{5\sqrt{3\sqrt{5\sqrt{...}}}}

.

Since the pattern repeats infinitely, we can write:

x = 3\sqrt{5\sqrt{x}}

Square both sides:

x^2 = 9 \cdot 5\sqrt{x}

x^2 = 45\sqrt{x}

Raise both sides to the power of 2 again (or square both sides after isolating the square root):

x^4 = 2025x

x^4 - 2025x = 0

x(x^3 - 2025) = 0

Since

x > 0

, we have:

x^3 = 2025

x = \sqrt[3]{2025}

Factor 2025:

2025 = 25 \cdot 81 = 5^2 \cdot 3^4

x = \sqrt[3]{5^2 \cdot 3^4} = \sqrt[3]{5^2 \cdot 3^3 \cdot 3} = 3\sqrt[3]{5^2 \cdot 3} = 3\sqrt[3]{75}

…

Comments

Be the first to share your thoughts and help others understand it better.

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.