Solved Examples

Example 1: Is 324 a perfect square?

Question: Determine if 324 is a perfect square using prime factorization.

Solution:

1. Factorize 324: $324 = 2 \times 162 = 2 \times 2 \times 81 = 2 \times 2 \times 9 \times 9 = 2 \times 2 \times 3 \times 3 \times 3 \times 3$
2. Group factors: $(2 \times 2) \times (3 \times 3) \times (3 \times 3)$
3. Since all factors can be paired completely with no remainders, Yes, 324 is a perfect square.
4. $\sqrt{324} = 2 \times 3 \times 3 = 18$.

Example 2: Is 156 a perfect square?

Question: Check 156.

Solution:

1. Factorize 156: $156 = 2 \times 78 = 2 \times 2 \times 39 = 2 \times 2 \times 3 \times 13$
2. Group factors: $(2 \times 2) \times 3 \times 13$
3. The factors 3 and 13 do not have pairs.
4. No, 156 is not a perfect square.

Example 3: Estimating $\sqrt{1936}$

Question: Find the square root of 1936 by estimation.

Solution:

1. Find Range: $40^2 = 1600$ and $50^2 = 2500$. 1936 is between them.
2. Check Unit Digit: 1936 ends in 6. The root must end in 4 or 6.
3. Possibilities: 44 or 46.
4. Narrow Down: $45^2 = 2025$.
5. Since $1936 < 2025$, the root must be less than 45.
6. Therefore, the root is 44.

Example 4: Is 3375 a perfect cube?

Question: Check if 3375 is a perfect cube.

Solution:

1. Factorize 3375: $3375 = 5 \times 675 = 5 \times 5 \times 135 = 5 \times 5 \times 5 \times 27$ $3375 = 5 \times 5 \times 5 \times 3 \times 3 \times 3$
2. Group in triplets: $(5 \times 5 \times 5) \times (3 \times 3 \times 3)$
3. All factors form triplets. Yes, it is a perfect cube.
4. $\sqrt[3]{3375} = 5 \times 3 = 15$.