Question 10

We will find the cube root of each number by breaking it down into its prime factors.

Step 1 — Prime Factorization of 27000

We divide 27000 by prime numbers until we reach 1. A prime number is a whole number greater than 1 with only two divisors: 1 and itself.

$27000 \div 2 = 13500$

$13500 \div 2 = 6750$

$6750 \div 2 = 3375$

$3375 \div 3 = 1125$

$1125 \div 3 = 375$

$375 \div 3 = 125$

$125 \div 5 = 25$

$25 \div 5 = 5$

$5 \div 5 = 1$

So, the prime factorization of 27000 is: $27000 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 5 \times 5 \times 5$

Step 2 — Finding the Cube Root of 27000

We group the prime factors into sets of three identical factors. The cube root is found by taking one factor from each group.

$\sqrt[3]{27000} = \sqrt[3]{(2 \times 2 \times 2) \times (3 \times 3 \times 3) \times (5 \times 5 \times 5)}$

$= 2 \times 3 \times 5$

$= 6 \times 5$

$\boxed{30}$

Step 3 — Prime Factorization of 10648

We divide 10648 by prime numbers until we reach 1.

$10648 \div 2 = 5324$

$5324 \div 2 = 2662$

$2662 \div 2 = 1331$

We check for divisibility by other prime numbers. We find that 1331 is divisible by 11.

$1331 \div 11 = 121$

$121 \div 11 = 11$

$11 \div 11 = 1$

So, the prime factorization of 10648 is: $10648 = 2 \times 2 \times 2 \times 11 \times 11 \times 11$

Step 4 — Finding the Cube Root of 10648

We group the prime factors into sets of three identical factors. The cube root is found by taking one factor from each group.

$\sqrt[3]{10648} = \sqrt[3]{(2 \times 2 \times 2) \times (11 \times 11 \times 11)}$

$= 2 \times 11$

$\boxed{22}$

Answer

(i) The cube root of 27000 is 30. (ii) The cube root of 10648 is 22.