Is 500 a perfect cube — if not find smallest number to multiply

Question

Is 500 a perfect cube? If not, find the smallest natural number by which 500 must be multiplied to make it a perfect cube. Also find the resulting cube root.

Solution — Step by Step

Using division by smallest prime factors:

500 = 2 \times 250 = 2 \times 2 \times 125 = 2^2 \times 5^3

So $500 = 2^2 \times 5^3$ .

A number is a perfect cube if and only if every prime factor in its factorization appears as a multiple of 3.

For $500 = 2^2 \times 5^3$ :

Power of 2: exponent is 2 → NOT a multiple of 3
Power of 5: exponent is 3 ✓

Since the exponent of 2 is 2 (not a multiple of 3), 500 is NOT a perfect cube.

To make 500 a perfect cube, we need the exponent of every prime to be a multiple of 3.

Currently: $2^2 \times 5^3$

For the factor $2^2$ : we need exponent to be 3 (next multiple of 3). We need to add $2^{3-2} = 2^1$ .
For the factor $5^3$ : already a multiple of 3. No additional 5s needed.

So we need to multiply by $2^1 = \mathbf{2}$ .

The smallest natural number to multiply 500 by is 2.

$500 \times 2 = 1000 = 2^3 \times 5^3 = (2 \times 5)^3 = 10^3$

$\sqrt[3]{1000} = 10$ ✓

The resulting perfect cube is 1000, and its cube root is 10.

Why This Works

A perfect cube has all prime factors appearing in groups of exactly 3. When we factorize and find that some primes appear in “incomplete groups” (exponents that aren’t multiples of 3), we need to multiply by whatever is missing to complete the groups.

For $2^2$ : we have 2 twos, but need 3 twos for a complete cube → multiply by one more 2. For $5^3$ : already complete → nothing needed.

This is the same logic used for perfect squares (all exponents must be even) but applied to cubes (all exponents must be multiples of 3).

Alternative Method

Think of it visually: a perfect cube can be arranged as a 3D cube of equal dimension. 500 unit cubes cannot form a perfect cube. 1000 = 10 × 10 × 10 unit cubes — that’s a perfect cube of side length 10. You need 2 more unit cubes to complete the arrangement (500 + 500… wait, $1000/500 = 2$ , so you needed 500 more to get there, but through multiplication: $500 \times 2 = 1000$ ).

The prime factorization method is far more rigorous and works for any number.

For CBSE Class 8 and similar exams, always show the complete prime factorization before checking. The method is: (1) factorize, (2) group prime factors into triples, (3) identify incomplete triples, (4) find what’s missing to complete each triple. For finding the smallest divisor (instead of multiplier) to make a perfect cube, the logic reverses: identify the “extra” factors that make exponents not multiples of 3 and divide by those.

Common Mistake

Students sometimes try to find the cube root directly without prime factorization: $\sqrt[3]{500} \approx 7.937$ — not a whole number, so 500 is not a perfect cube. This confirms it’s not a perfect cube, but it doesn’t tell you what to multiply by. You need the prime factorization method to find the multiplier. Another common error: after finding the multiplier is 2, some students compute $\sqrt[3]{500 \times 2} = \sqrt[3]{1000}$ but then forget to simplify $\sqrt[3]{1000} = 10$ . Always complete the calculation to find the actual cube root.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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