27000 = 2³ × 3³ × 5³ → ∛27000 = 2 × 3 × 5 = 30.

Find Cube Root of 27000 by Prime Factorisation

Question

Find the cube root of 27000 using the prime factorisation method.

Solution — Step by Step

We break 27000 down completely into its prime factors. Start by dividing by the smallest primes:

27000 = 2 \times 13500 = 2 \times 2 \times 6750 = 2 \times 2 \times 2 \times 3375

3375 = 3 \times 1125 = 3 \times 3 \times 375 = 3 \times 3 \times 3 \times 125

125 = 5 \times 25 = 5 \times 5 \times 5

So the complete factorisation is: $27000 = 2^3 \times 3^3 \times 5^3$

For cube roots, we need groups of three identical factors. Here, every prime factor already forms a perfect triplet — three 2s, three 3s, three 5s. This tells us 27000 is a perfect cube, so the cube root will be a whole number.

This is the key idea: $\sqrt[3]{a^3} = a$ . So we take one factor from each group of three:

\sqrt[3]{27000} = \sqrt[3]{2^3 \times 3^3 \times 5^3} = 2 \times 3 \times 5

2 \times 3 \times 5 = \mathbf{30}

\therefore \sqrt[3]{27000} = \mathbf{30}

Quick verification: $30^3 = 30 \times 30 \times 30 = 900 \times 30 = 27000$ ✓

Why This Works

The cube root undoes the cubing operation. When we write $27000 = 2^3 \times 3^3 \times 5^3$ , we can use the rule $\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}$ to split the cube root across each prime group.

Each triplet $p^3$ gives exactly $p$ when we take its cube root. So the whole problem reduces to: find the triplets, pick one from each, multiply. No guesswork needed.

This method works for any perfect cube. If a number is not a perfect cube, you’ll find at least one prime factor that doesn’t form a complete group of three — the method tells you that too.

Alternative Method

Using the shortcut: split 27000 as a product of known cubes.

27000 = 27 \times 1000 = 3^3 \times 10^3

\sqrt[3]{27000} = \sqrt[3]{27} \times \sqrt[3]{1000} = 3 \times 10 = \mathbf{30}

This is faster in exams once you recognise that $27 = 3^3$ and $1000 = 10^3$ . In NCERT Class 8 exams, numbers are often chosen so they split cleanly like this — worth training your eye to spot these pairs.

Memorise cubes up to 15: $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375$ . In most board questions, the answer is one of these, and spotting the split saves you the full factorisation.

Common Mistake

Stopping the factorisation too early. Many students write $27000 = 27 \times 1000$ and then factorise only $27 = 3^3$ , forgetting to also factorise 1000 down to $2^3 \times 5^3$ . They then write $\sqrt[3]{27000} = 3 \times \sqrt[3]{1000}$ and get stuck. Always factorise every part completely before grouping — or use the shortcut only when you’re confident about both factors.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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