1. Find the cube root of each of the following numbers by prime factorisation method.
(i) 64 (ii) 512 (iii) 10648 (iv) 27000 (v) 15625
(vi) 13824 (vii) 110592 (viii) 46656 (ix) 175616 (x) 91125
(i) 64
Step 1: Prime factorization of 64
| Divisor | Number |
|---|---|
| 2 | 64 |
| 2 | 32 |
| 2 | 16 |
| 2 | 8 |
| 2 | 4 |
| 2 | 2 |
| 1 |
Step 2: Prime factorization expression
\[64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \]
\[ 2^3 \times 2^3 \]
\[ 2 \times 2 =4 \]
∴ Cube root of 64 is 4.
(ii) 512
Step 1: Prime factorization of 512
| Divisor | Number |
|---|---|
| 2 | 512 |
| 2 | 256 |
| 2 | 128 |
| 2 | 64 |
| 2 | 32 |
| 2 | 16 |
| 2 | 8 |
| 2 | 4 |
| 2 | 2 |
| 1 |
Step 2: Prime factorization expression
\[512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \]
\[ 2^3 \times 2^3 \times 2^3 \]
\[ 2 \times 2 \times 2 = 8 \]
∴ Cube root of 512 is 8.
(iii) 10648
Step 1: Prime factorization of 10648
| Divisor | Number |
|---|---|
| 2 | 10648 |
| 2 | 5324 |
| 2 | 2662 |
| 11 | 1331 |
| 11 | 121 |
| 11 | 11 |
| 1 |
Step 2: Prime factorization expression
\[10648 = 2 \times 2 \times 2 \times 11 \times 11 \times 11 \]
\[= 2^3 \times 11^3\]
\[= 2 \times 11 = 22\]
∴ Cube root of 10648 is 22.
(iv) 27000
Step 1: Prime factorization of 27000
| Divisor | Number |
|---|---|
| 2 | 27000 |
| 2 | 13500 |
| 2 | 6750 |
| 3 | 3375 |
| 3 | 1125 |
| 3 | 375 |
| 5 | 125 |
| 5 | 25 |
| 5 | 5 |
| 1 |
Step 2: Prime factorization expression
\[27000 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 5 \times 5 \times 5 \]
\[= 2^3 \times 3^3 \times 5^3\]
\[= 2 \times 3 \times 5 = 30\]
∴ Cube root of 27000 is 30.
(v) 15625
Step 1: Prime factorization of 15625
| Divisor | Number |
|---|---|
| 5 | 15625 |
| 5 | 3125 |
| 5 | 625 |
| 5 | 125 |
| 5 | 25 |
| 5 | 5 |
| 1 |
Step 2: Prime factorization expression
\[15625 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \]
\[= 5^3 \times 5^3\]
\[= 5 \times 5 = 25\]
∴ Cube root of 15625 is 25.
2. State true or false.
(i) Cube of any odd number is even.
"False". The cube of an odd number is always odd.
Example: 3³ = 27 (odd), 5³ = 125 (odd)
(ii) A perfect cube does not end with two zeros.
"True". A perfect cube ending with two zeros must end with at least three zeros.
Example: 10³ = 1000, 20³ = 8000
(iii) If square of a number ends with 5, then its cube ends with 25.
False".. The cube of a number ending with 5 always ends with 125, not 25.
Example: 5² = 25, 5³ = 125; 15² = 225, 15³ = 3375
(iv) There is no perfect cube which ends with 8.
False".. There are perfect cubes that end with 8.
Example: 2³ = 8, 12³ = 1728, 22³ = 10648
(v) The cube of a two digit number may be a three digit number.
False".. The smallest two-digit number is 10, and 10³ = 1000 (four digits).
The cube of any two-digit number will have at least four digits.
(vi) The cube of a two digit number may have seven or more digits.
False".. The largest two-digit number is 99, and 99³ = 970299 (six digits).
The cube of any two-digit number will have at most six digits.
(vii) The cube of a single digit number may be a single digit number.
"True". The cube of 1 is 1, and the cube of 2 is 8, both are single-digit numbers.