A number has the prime factorisation 2⁶ × 5³ × 7³. Find its cube root.

140. Divide each exponent by 3: 2⁶ contributes 2² = 4, 5³ contributes 5 and 7³ contributes 7. The cube root is 4 × 5 × 7 = 140. Picking 70 takes only one 2 from 2⁶, while 280 halves the exponent (2³ = 8) as if taking a square root; 350 wrongly keeps 5².

A number has the prime factorisation $2^{6} \times 5^{3} \times 7^{3}$ . Find its cube root.

Class 8Cubes and Cube RootsCubes and Roots

A70
B140
C280
D350

Step-by-Step Solution

Divide each exponent by 3:

2^{6}

contributes

2^{2} = 4, 5^{3}

contributes 5 and

7^{3}

contributes 7. The cube root is

4 \times 5 \times 7 = 140

. Picking 70 takes only one 2 from

2^{6}

, while 280 halves the exponent

(2^{3} = 8)

as if taking a square root; 350 wrongly keeps

5^{2}

Answer: B) 140

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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Step-by-Step Solution

Key idea: Cube root by prime factorisation

Master this chapter in 30 minutes a day.

More Class 8 Cubes and Cube Roots practice

A number has the prime factorisation $2^{6} \times 5^{3} \times 7^{3}$ . Find its cube root.