500 = 2² × 5³. Need one more 2 → multiply by 2.

Is 500 a Perfect Cube? Check and Find What to Multiply

Question

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

Solution — Step by Step

We break 500 down to its prime factors:

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

Always write prime factors in exponential form — it makes the next step much cleaner.

For a perfect cube, every prime factor must appear in groups of 3 (i.e., exponent divisible by 3).

500 = 2^2 \times 5^3

The 5’s are fine — they form one complete triplet. The 2’s only appear twice, so we have an incomplete triplet.

We need the exponent of each prime to be a multiple of 3.

For 5: exponent is 3 → already complete ✓
For 2: exponent is 2 → needs one more 2 to reach 3

So we must multiply by 2¹ = 2.

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

1000 is a perfect cube, and its cube root is 10.

The smallest number to multiply 500 by is 2, giving us 1000, which is $10^3$ .

Why This Works

A perfect cube means the number can be written as $n^3$ for some integer $n$ . When we write a number as a product of primes, each prime’s exponent must be exactly divisible by 3 — because $n^3$ means every prime in $n$ appears three times as often.

Think of it like arranging objects in a 3D cube. If you have 8 objects, you can form a $2 \times 2 \times 2$ cube. If you have 9, you can’t — you’re one short of the next perfect cube ( $3 \times 3 \times 3 = 27$ ). The prime factorisation method makes this shortfall precise and calculable.

The key insight is that we only need to find what’s missing from each triplet, not rebuild the whole number. This is why prime factorisation is so powerful here — it turns a multiplication problem into simple counting.

Alternative Method

You can use the cube root estimation approach to verify, though prime factorisation is always preferred in exams.

We know $8^3 = 512$ and $7^3 = 343$ . Since $343 < 500 < 512$ , the cube root of 500 lies between 7 and 8 — so 500 is clearly not a perfect cube.

But this method only confirms 500 isn’t a perfect cube; it doesn’t tell us what to multiply by. For the full answer, we must use prime factorisation.

In CBSE Class 8 exams, “find the smallest number to multiply/divide to make a perfect cube” is a guaranteed 2-3 mark question every year. Memorise the method: prime factorise → group into triplets → fill the gap.

Common Mistake

Students often confuse “multiply by 2” with “multiply by 4”. Here’s why: they see $2^2$ in the factorisation and think they need two more 2’s to complete another triplet — writing $2^2 \times 2^2 = 2^4$ . But that’s wrong. We already have two 2’s; we only need one more to complete the current triplet of three. The target is $2^3$ , not $2^6$ .

Question

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

Solution — Step by Step

We break 500 down to its prime factors:

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

Always write prime factors in exponential form — it makes the next step much cleaner.

For a perfect cube, every prime factor must appear in groups of 3 (i.e., exponent divisible by 3).

500 = 2^2 \times 5^3

The 5’s are fine — they form one complete triplet. The 2’s only appear twice, so we have an incomplete triplet.

We need the exponent of each prime to be a multiple of 3.

For 5: exponent is 3 → already complete ✓
For 2: exponent is 2 → needs one more 2 to reach 3

So we must multiply by 2¹ = 2.

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

1000 is a perfect cube, and its cube root is 10.

The smallest number to multiply 500 by is 2, giving us 1000, which is $10^3$ .

Why This Works

Alternative Method

You can use the cube root estimation approach to verify, though prime factorisation is always preferred in exams.

We know $8^3 = 512$ and $7^3 = 343$ . Since $343 < 500 < 512$ , the cube root of 500 lies between 7 and 8 — so 500 is clearly not a perfect cube.

But this method only confirms 500 isn’t a perfect cube; it doesn’t tell us what to multiply by. For the full answer, we must use prime factorisation.

In CBSE Class 8 exams, “find the smallest number to multiply/divide to make a perfect cube” is a guaranteed 2–3 mark question every year. Memorise the method: prime factorise → group into triplets → fill the gap.

Common Mistake

Students often multiply by 4 instead of 2. They see $2^2$ in the factorisation and think they need two more 2’s to complete another full triplet — but we already have two 2’s sitting there. We only need one more to complete the current triplet of three. The target is $2^3$ , not starting fresh at $2^6$ .

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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