Find cube root of 27000 using prime factorization

Question

Find the cube root of 27,000 using the prime factorization method.

(NCERT Class 8)

Solution — Step by Step

Divide step by step:

$27000 = 27 \times 1000$

$27 = 3 \times 3 \times 3 = 3^3$

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

So: $27000 = 3^3 \times 2^3 \times 5^3$

For a perfect cube, every prime factor must appear in groups of three:

27000 = 3^3 \times 2^3 \times 5^3

Each prime (3, 2, and 5) appears exactly 3 times. So 27,000 is a perfect cube.

Take one factor from each triplet:

\sqrt[3]{27000} = 3 \times 2 \times 5 = \mathbf{30}

Verification: $30^3 = 30 \times 30 \times 30 = 900 \times 30 = 27000$ ✓

Why This Works

The cube root “undoes” the cubing operation. Since cubing means multiplying a number by itself three times, the prime factorization of a perfect cube will always have every prime appearing in multiples of three. Taking one from each group of three reverses the process.

If any prime didn’t appear in a multiple of three, the number would NOT be a perfect cube, and we couldn’t find an integer cube root.

Alternative Method — Breaking into Known Cubes

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

This works because $\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}$ , and we already know that $\sqrt[3]{27} = 3$ and $\sqrt[3]{1000} = 10$ .

Memorise cubes of 1 to 15: $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375$ . This helps you spot perfect cubes quickly and saves time in exams. For larger numbers, use prime factorization.

Common Mistake

Students sometimes confuse cube root with dividing by 3. The cube root of 27000 is NOT $27000/3 = 9000$ . Cube root means finding a number that, when multiplied by itself three times, gives 27000. $30 \times 30 \times 30 = 27000$ , so $\sqrt[3]{27000} = 30$ .

Question

Solution — Step by Step

Why This Works

Alternative Method — Breaking into Known Cubes

Common Mistake

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