Finding Square Roots through Prime Factorization
To find the square root of a number using prime factorization, we first express the number as a product of its prime factors. For a perfect square, each prime factor will occur an even number of times. For example, considering the number 36:
- Prime factorization of 36 is:
36 = 2 × 2 × 3 × 3. Each prime factor (2 and 3) occurs twice.
When we find the square root of 36, we group the prime factors in pairs:
t = (2 × 3)².
Thus, the square root of 36 is 6.
In contrast, if we analyze a number like 90,
t = 90 = 2 × 3 × 3 × 5,
it is not a perfect square since 2 and 5 do not have pairs. We learn that to form a perfect square, we can multiply the number by the missing factors necessary to complete the pairs.
The significance of this method lies in its effectiveness for larger numbers or identifying perfect squares, which is useful in various mathematical applications.
Similar Questions
- Example: Is 1440 a perfect square? If not, find the smallest multiple of 1440 that is a perfect square. Find the square root of the new number.
Solution: We have 1440 = 2^5 \times 3^2 \times 5^1.
The prime factor 5 has no pair, thus 1440 is not a perfect square. So, we multiply 1440 by 5 to get,
1440 \times 5 = 7200 = 2^5 \times 3^2 \times 5^2.
Now each prime factor has a pair. Therefore, 7200 is a perfect square.
Thus, the required smallest multiple of 1440 which is a perfect square is 7200, with a square root of 84.
\( \sqrt{7200} = 60 \)
- Example 8: Is 5000 a perfect square? If not, find the smallest multiple of 5000 which is a perfect square. Find the square root of this new number.
Solution: We have 5000 = 2^3 \times 5^4.
The prime factor 2 has no pair, thus 5000 is not a perfect square. If we multiply by 2, we get 10000 = 2^4 \times 5^4, which is a perfect square.
Thus, the required number is \( 10000 \), with a square root of \( \sqrt{10000} = 100 \).
- Example 9: Is 2500 a perfect square? If not, find the smallest multiple of 2500 that is a perfect square. Then, find the square root of this new number.
Solution: We have 2500 = 2^2 \times 5^4.
Since all prime factors have pairs, 2500 is a perfect square itself. The square root is \( \sqrt{2500} = 50 \).
- Example 10: Is 980 a perfect square? If not, determine the smallest multiple of 980 that becomes a perfect square, and find the square root of this new number.
Solution: We have 980 = 2^2 \times 5^1 \times 7^2.
The prime factor 5 has no pair, thus 980 is not a perfect square. By multiplying by 5, we get 4900 = 2^2 \times 5^2 \times 7^2 which is perfect.
Hence, the smallest multiple of 980 which is a perfect square is 4900, giving a square root of \( \sqrt{4900} = 70 \).
- Example 11: Is 850 a perfect square? If not, locate the smallest multiple of 850 which is a perfect square, and compute the square root of this new number.
Solution: We have 850 = 2^1 \times 5^2 \times 17^1.
The prime factors 2 and 17 have no pairs, thus 850 is not a perfect square. Multiplying by 34 results in 28900 = 2^2 \times 5^2 \times 17^2, which is a perfect square. Therefore, the smallest multiple is 28900, with a square root of \( \sqrt{28900} = 170 \).