What is the unit digit in the cube roots of the following perfect cubes? a) 4096 b) 6859 c) 19683 d) 140608 e) -1520875. 2. Which of the following are cubes of odd natural numbers? a) 6859 b) 4913

Here are the solutions to the problems.

1. What is the unit digit in the cube roots of the following perfect cubes?
To find the unit digit of a cube root, we look at the unit digit of the number itself.

• a) 4096

  Step 1: The unit digit of 4096 is 6.

  Step 2: Since $6^3 = 216$, the unit digit of the cube root will be 6.
  The unit digit is $\boxed{6}$.

• b) 6859

  Step 1: The unit digit of 6859 is 9.

  Step 2: Since $9^3 = 729$, the unit digit of the cube root will be 9.
  The unit digit is $\boxed{9}$.

• c) 19683

  Step 1: The unit digit of 19683 is 3.

  Step 2: Since $7^3 = 343$, the unit digit of the cube root will be 7.
  The unit digit is $\boxed{7}$.

• d) 140608

  Step 1: The unit digit of 140608 is 8.

  Step 2: Since $2^3 = 8$, the unit digit of the cube root will be 2.
  The unit digit is $\boxed{2}$.

• e) -1520875

  Step 1: The unit digit of -1520875 is 5.

  Step 2: Since $5^3 = 125$, the unit digit of the cube root will be 5.
  The unit digit is $\boxed{5}$.

2. Which of the following are cubes of odd natural numbers?
A cube of an odd natural number is always an odd number.

• a) 6859

  Step 1: The number 6859 is odd (ends in 9).

  Step 2: Find the cube root: $\sqrt[3]{6859} = 19$.

  Step 3: Since 19 is an odd natural number, 6859 is the cube of an odd natural number.
  $\boxed{\text{Yes}}$

• b) 4913

  Step 1: The number 4913 is odd (ends in 3).

  Step 2: Find the cube root: $\sqrt[3]{4913} = 17$.

  Step 3: Since 17 is an odd natural number, 4913 is the cube of an odd natural number.
  $\boxed{\text{Yes}}$

• c) 10648

  Step 1: The number 10648 is even (ends in 8).

  Step 2: Find the cube root: $\sqrt[3]{10648} = 22$.

  Step 3: Since 22 is an even natural number, 10648 is not the cube of an odd natural number.
  $\boxed{\text{No}}$

• d) 2197

  Step 1: The number 2197 is odd (ends in 7).

  Step 2: Find the cube root: $\sqrt[3]{2197} = 13$.

  Step 3: Since 13 is an odd natural number, 2197 is the cube of an odd natural number.
  $\boxed{\text{Yes}}$

3. Which of the following are cubes of even natural numbers?
A cube of an even natural number is always an even number.

• a) 13824

  Step 1: The number 13824 is even (ends in 4).

  Step 2: Find the cube root: $\sqrt[3]{13824} = 24$.

  Step 3: Since 24 is an even natural number, 13824 is the cube of an even natural number.
  $\boxed{\text{Yes}}$

• b) 42875

  Step 1: The number 42875 is odd (ends in 5).

  Step 2: Find the cube root: $\sqrt[3]{42875} = 35$.

  Step 3: Since 35 is an odd natural number, 42875 is not the cube of an even natural number.
  $\boxed{\text{No}}$

• c) 8000

  Step 1: The number 8000 is even (ends in 0).

  Step 2: Find the cube root: $\sqrt[3]{8000} = 20$.

  Step 3: Since 20 is an even natural number, 8000 is the cube of an even natural number.
  $\boxed{\text{Yes}}$

• d) 936056

  Step 1: The number 936056 is even (ends in 6).

  Step 2: Find the cube root: $\sqrt[3]{936056} = 98$.

  Step 3: Since 98 is an even natural number, 936056 is the cube of an even natural number.
  $\boxed{\text{Yes}}$

4. Find the cube roots of the following by prime factorisation.

• a) 512

  Step 1: Prime factorise 512.

  $$512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^9$$

  Step 2: Group factors in threes.

  $$512 = (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2) = 2^3 \times 2^3 \times 2^3 = (2 \times 2 \times 2)^3 = 8^3$$

  Step 3: Find the cube root.

  $$\sqrt[3]{512} = \sqrt[3]{8^3} = \boxed{8}$$

• b) 4096

  Step 1: Prime factorise 4096.

  $$4096 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^{12}$$

  Step 2: Group factors in threes.

  $$4096 = (2^3) \times (2^3) \times (2^3) \times (2^3) = (2 \times 2 \times 2 \times 2)^3 = 16^3$$

  Step 3: Find the cube root.

  $$\sqrt[3]{4096} = \sqrt[3]{16^3} = \boxed{16}$$

• c) 15625

  Step 1: Prime factorise 15625.

  $$15625 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 5^6$$

  Step 2: Group factors in threes.

  $$15625 = (5 \times 5) \times (5 \times 5) \times (5 \times 5) = (5^2)^3 = 25^3$$

  Step 3: Find the cube root.

  $$\sqrt[3]{15625} = \sqrt[3]{25^3} = \boxed{25}$$

• d) 27000

  Step 1: Prime factorise 27000.

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