The square root of a natural number is n times its cubic root. That number is -

Concept

For any positive real number written as a power, xa ÷ xb = xa−b. The square root is the exponent 1/2 and the cube root is the exponent 1/3, so dividing one root by the other subtracts their exponents.

Application

1. Let the natural number be x. The condition states √x = n · ∛x.

2. Write the roots as powers: x1/2 = n · x1/3.

3. Divide both sides by x1/3: x1/2 ÷ x1/3 = n.

4. Subtract the exponents: 1/2 − 1/3 = 3/6 − 2/6 = 1/6, so x1/6 = n.

5. Raise both sides to the 6th power: x = n6.

Cross-check

Substitute x = n6: √(n6) = n3 and ∛(n6) = n2. Their ratio n3 ÷ n2 = n, which matches the required factor of n.