Ravi multiplies 250 by the smallest whole number that turns it into a perfect cube. What product does he get?

Class 8Cubes and Cube RootsCubes and Roots

A500
B1000
C2000
D8000

Step-by-Step Solution

250 = 2 ×

5^{3}

. The

5^{3}

is complete, but the lone 2 needs two more 2s, so multiply by

2^{2} = 4 : 250 \times 4 = 1000 = 1 0^{3}

. Doubling to

500 = 2^{2} \times 5^{3}

still leaves the 2s short of a triple.

Answer: B) 1000

Key idea: Cube root by prime factorisation

**Step 1**: factorise into primes.
**Step 2**: group the prime factors into TRIPLES (three of the same).
**Step 3**: take ONE factor from each triple — their product is the cube root.

Example: ∛216.

216 = 2^{3} \times 3^{3} = (2 \times 3)^{3} = 6^{3}

. So ∛216 = 6.

Example: ∛13824.

13824 = 2^{9} \times 3^{3} = (2^{3} \times 3)^{3} = (8 \times 3)^{3} = 2 4^{3}

. So ∛13824 = 24.

If any prime doesn't form a complete triple, the number isn't a perfect cube.

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