Answer :
Let's go through the questions step by step:
- Find the cube of the following numbers:
(a) 13:
The cube of 13 is given by [tex]13^3 = 13 \times 13 \times 13 = 2197[/tex].
(b) 66:
The cube of 66 is [tex]66^3 = 66 \times 66 \times 66 = 287496[/tex].
(c) -15:
The cube of -15 is [tex](-15)^3 = (-15) \times (-15) \times (-15) = -3375[/tex].
(d) 5.8:
The cube of 5.8 is [tex]5.8^3 = 5.8 \times 5.8 \times 5.8 \approx 195.112[/tex].
(e) -10.1:
The cube of -10.1 is [tex](-10.1)^3 = (-10.1) \times (-10.1) \times (-10.1) \approx -1030.301[/tex].
(f) [tex]\frac{3}{14}[/tex]:
The cube of [tex]\frac{3}{14}[/tex] is [tex]\left(\frac{3}{14}\right)^3 = \frac{3 \times 3 \times 3}{14 \times 14 \times 14} = \frac{27}{2744}[/tex].
(g) [tex]-\frac{5}{11}[/tex]:
The cube of [tex]-\frac{5}{11}[/tex] is [tex]\left(-\frac{5}{11}\right)^3 = \frac{-5 \times -5 \times -5}{11 \times 11 \times 11} = \frac{-125}{1331}[/tex].
- Check whether the given numbers are perfect cubes or not:
To determine if a number is a perfect cube, we check if its cube root is an integer.
(a) 343:
The cube root of 343 is 7, which is an integer, so 343 is a perfect cube.
(b) 625:
The cube root of 625 is approximately 8.5499, which is not an integer, so 625 is not a perfect cube.
(c) 1728:
The cube root of 1728 is 12, which is an integer, so 1728 is a perfect cube.
(d) 2744:
The cube root of 2744 is 14, which is an integer, so 2744 is a perfect cube.
(e) 6589:
The cube root of 6589 is approximately 18.772, which is not an integer, so 6589 is not a perfect cube.
(f) 91125:
The cube root of 91125 is approximately 45.024, which is not an integer, so 91125 is not a perfect cube.
(g) 27000:
The cube root of 27000 is 30, which is an integer, so 27000 is a perfect cube.
- Identify the numbers whose cubes are even numbers:
An even number raised to any power results in an even number. Thus, we look for even numbers in the list.
(a) 122 - Even
(c) 728 - Even
(d) 2300 - Even
(g) 9000 - Even
- Identify the numbers whose cubes are odd numbers:
An odd number raised to any power results in an odd number. Thus, we look for odd numbers in the list.
(a) 55 - Odd
(c) 1227 - Odd
(e) 9813 - Odd
- Two examples to show that the cube of an even number is always even:
- Example 1: [tex]2^3 = 2 \times 2 \times 2 = 8[/tex], which is even.
- Example 2: [tex]4^3 = 4 \times 4 \times 4 = 64[/tex], which is even.
- Two examples to show that the cube of an odd number is always odd:
- Example 1: [tex]3^3 = 3 \times 3 \times 3 = 27[/tex], which is odd.
- Example 2: [tex]5^3 = 5 \times 5 \times 5 = 125[/tex], which is odd.
