Squares and Square Roots, Cube and Cube Roots

Q1: Find the cubes of:

i. \(16^3\)

Step 1: Cube 16: \[ 16^3 = 16 \times 16 \times 16 = 4096 \]Answer: \(16^3 = 4096\)

ii. \(30^3\)

Step 1: Cube 30: \[ 30^3 = 30 \times 30 \times 30 = 27000 \]Answer: \(30^3 = 27000\)

iii. \(1.2^3\)

Step 1: Cube 1.2: \[ 1.2^3 = 1.2 \times 1.2 \times 1.2 = 1.728 \]Answer: \(1.2^3 = 1.728\)

iv. \(0.7^3\)

Step 1: Cube 0.7: \[ 0.7^3 = 0.7 \times 0.7 \times 0.7 = 0.343 \]Answer: \(0.7^3 = 0.343\)

v. \(0.06^3\)

Step 1: Cube 0.06: \[ 0.06^3 = 0.06 \times 0.06 \times 0.06 = 0.000216 \]Answer: \(0.06^3 = 0.000216\)

vi. \(\left(\frac{2}{5}\right)^3\)

Step 1: Cube \(\frac{2}{5}\): \[ \left(\frac{2}{5}\right)^3 = \frac{2^3}{5^3} = \frac{8}{125} \]Answer: \(\left(\frac{2}{5}\right)^3 = \frac{8}{125}\)

vii. \(\left(\frac{1}{9}\right)^3\)

Step 1: Cube \(\frac{1}{9}\): \[ \left(\frac{1}{9}\right)^3 = \frac{1^3}{9^3} = \frac{1}{729} \]Answer: \(\left(\frac{1}{9}\right)^3 = \frac{1}{729}\)

viii. \(\left(1\frac{3}{5}\right)^3\)

Step 1: Convert the mixed number to an improper fraction: \[ 1\frac{3}{5} = \frac{8}{5} \]Step 2: Cube \(\frac{8}{5}\): \[ \left(\frac{8}{5}\right)^3 = \frac{8^3}{5^3} = \frac{512}{125} \]Answer: \(\left(1\frac{3}{5}\right)^3 = \frac{512}{125}\)

Q2: Examine which of the following numbers are perfect cubes. In case of perfect cube, find the cube root:

i. \(729\)

Step 1: Check if 729 is a perfect cube: \[ \sqrt[3]{729} = 9 \quad \text{(since \(9^3 = 729\))} \]Answer: \(729\) is a perfect cube, and the cube root is \(9\)

ii. \(1331\)

Step 1: Check if 1331 is a perfect cube: \[ \sqrt[3]{1331} = 11 \quad \text{(since \(11^3 = 1331\))} \]Answer: \(1331\) is a perfect cube, and the cube root is \(11\)

iii. \(5324\)

Step 1: Check if 5324 is a perfect cube: \[ \sqrt[3]{5324} \approx 17.2 \quad \text{(not an integer, hence not a perfect cube)} \]Answer: \(5324\) is not a perfect cube

iv. \(3375\)

Step 1: Check if 3375 is a perfect cube: \[ \sqrt[3]{3375} = 15 \quad \text{(since \(15^3 = 3375\))} \]Answer: \(3375\) is a perfect cube, and the cube root is \(15\)

v. \(9261\)

Step 1: Check if 9261 is a perfect cube: \[ \sqrt[3]{9261} = 21 \quad \text{(since \(21^3 = 9261\))} \]Answer: \(9261\) is a perfect cube, and the cube root is \(21\)

vi. \(5832\)

Step 1: Check if 5832 is a perfect cube: \[ \sqrt[3]{5832} = 18 \quad \text{(since \(18^3 = 5832\))} \]Answer: \(5832\) is a perfect cube, and the cube root is \(18\)

vii. \(1728\)

Step 1: Check if 1728 is a perfect cube: \[ \sqrt[3]{1728} = 12 \quad \text{(since \(12^3 = 1728\))} \]Answer: \(1728\) is a perfect cube, and the cube root is \(12\)

viii. \(10584\)

Step 1: Check if 10584 is a perfect cube: \[ \sqrt[3]{10584} \approx 21.5 \quad \text{(not an integer, hence not a perfect cube)} \]Answer: \(10584\) is not a perfect cube

Q3: Find the smallest by which 1323 must multiplied so that the product is a perfect cube.

Step 1: First, find the prime factorization of 1323: \[ 1323 \div 3 = 441 \] \[ 441 \div 3 = 147 \] \[ 147 \div 3 = 49 \] \[ 49 = 7 \times 7 \] Thus, the prime factorization of 1323 is: \[ 1323 = 3^3 \times 7^2 \]Step 2: To make 1323 a perfect cube, all exponents must be multiples of 3. Currently, the exponent of 7 is 2, which is not a multiple of 3. We need to multiply by one more 7 to make the exponent of 7 equal to 3.
Step 3: Multiply by 7 to make the exponent of 7 equal to 3: \[ 1323 \times 7 = 3^3 \times 7^3 \] Now, the product is a perfect cube.
Answer: The smallest number by which 1323 must be multiplied is \(7\), making the product \(1323 \times 7 = 9261\), which is a perfect cube.

Q4: What is the smallest number by which 1600 must be divided so that the quotient is a perfect cube?

Step 1: First, find the prime factorization of 1600: \[ 1600 \div 2 = 800 \] \[ 800 \div 2 = 400 \] \[ 400 \div 2 = 200 \] \[ 200 \div 2 = 100 \] \[ 100 \div 2 = 50 \] \[ 50 \div 2 = 25 \] \[ 25 \div 5 = 5 \] \[ 5 \div 5 = 1 \] Thus, the prime factorization of 1600 is: \[ 1600 = 2^6 \times 5^2 \]Step 2: To make the quotient a perfect cube, all exponents must be multiples of 3. The exponent of 2 is 6, which is already divisible by 3. However, the exponent of 5 is 2, which is not divisible by 3. We need to divide by one more 5 to make the exponent of 5 equal to 3.
Step 3: Divide by 5 to make the exponent of 5 equal to 3: \[ \frac{1600}{25} = 2^6 \] Now, the quotient is a perfect cube.
Answer: The smallest number by which 1600 must be divided is \(25\), making the quotient \(1600 \div 25 = 64\), which is a perfect cube.

Q5: Find the smallest number by which 2560 must be multiplied so that the product is a perfect cube.

Step 1: First, find the prime factorization of 2560: \[ 2560 \div 2 = 1280 \] \[ 1280 \div 2 = 640 \] \[ 640 \div 2 = 320 \] \[ 320 \div 2 = 160 \] \[ 160 \div 2 = 80 \] \[ 80 \div 2 = 40 \] \[ 40 \div 2 = 20 \] \[ 20 \div 2 = 10 \] \[ 10 \div 2 = 5 \] \[ 5 \div 5 = 1 \] Thus, the prime factorization of 2560 is: \[ 2560 = 2^9 \times 5 \]Step 2: To make 2560 a perfect cube, all exponents must be multiples of 3.
– The exponent of 2 is \(9\), which is divisible by 3.
– The exponent of 5 is \(1\), which is not divisible by 3. We need two more factors of 5 to make the exponent of 5 equal to 3.
Step 3: Multiply by \(5\times 5 = 25\) to make the exponents of 5 divisible by 3: \[ 2560 \times 25 = 2^9 \times 5^3 \] Now, the product is a perfect cube.
Answer: The smallest number by which 2560 must be multiplied is \(25\), making the product \(2560 \times 25 = 64000\), which is a perfect cube.

Q6: Find the cube root of:

i. \(\frac{216}{2197}\)

Step 1: Find the cube root of the numerator and denominator separately: \[ \sqrt[3]{216} = 6 \quad \text{(since \(6^3 = 216\))} \] \[ \sqrt[3]{2197} = 13 \quad \text{(since \(13^3 = 2197\))} \]Step 2: Now take the cube root of the fraction: \[ \sqrt[3]{\frac{216}{2197}} = \frac{\sqrt[3]{216}}{\sqrt[3]{2197}} = \frac{6}{13} \]Answer: The cube root of \(\frac{216}{2197}\) is \(\frac{6}{13}\)

ii. \(4\frac{508}{1331}\)

Step 1: Convert the mixed number into an improper fraction: \[ 4\frac{508}{1331} = \frac{4 \times 1331 + 508}{1331} = \frac{5324 + 508}{1331} = \frac{5832}{1331} \]Step 2: Find the cube root of the numerator and denominator separately: \[ \sqrt[3]{5832} = 18 \quad \text{(since \(18^3 = 5832\))} \] \[ \sqrt[3]{1331} = 11 \quad \text{(since \(11^3 = 1331\))} \]Step 3: Now take the cube root of the fraction: \[ \sqrt[3]{\frac{5832}{1331}} = \frac{\sqrt[3]{5832}}{\sqrt[3]{1331}} = \frac{18}{11} \]Answer: The cube root of \(4\frac{508}{1331}\) is \(\frac{18}{11}\)

iii. 42.875

Step 1: Express 42.875 as a fraction: \[ 42.875 = \frac{42875}{1000} \]Step 2: Find the cube root of the numerator and denominator separately: \[ \sqrt[3]{42875} = 35 \quad \text{(since \(35^3 = 42875\))} \] \[ \sqrt[3]{1000} = 10 \quad \text{(since \(10^3 = 1000\))} \]Step 3: Now take the cube root of the fraction: \[ \sqrt[3]{42.875} = \frac{35}{10} = 3.5 \]Answer: The cube root of 42.875 is \(3.5\)