Is 300 a perfect square? If not, find the smallest multiple whose square root is the number obtained. Find the smallest number by which 1,458 should be multiplied to get a perfect square. Also, find the square root of the perfect square so obtained.
The Correct Answer and Explanation is:
Let’s break down the question step by step.
1. Is 300 a perfect square?
A perfect square is a number that can be expressed as the square of an integer. In other words, if the square root of a number is an integer, then the number is a perfect square.
The square root of 300 is approximately:300≈17.32\sqrt{300} \approx 17.32300≈17.32
Since this is not an integer, 300 is not a perfect square.
2. Find the smallest multiple whose square root is the number obtained.
To find the smallest multiple of 300 that is a perfect square, we need to identify the prime factorization of 300 and adjust it to make sure that all exponents of the prime factors are even (since a perfect square requires all prime factors to appear with an even exponent).
Prime factorization of 300:300=22×3×52300 = 2^2 \times 3 \times 5^2300=22×3×52
For a perfect square, the exponents of all prime factors must be even. Here, the exponent of 3 is odd, so we need to multiply by 3 to make the exponent of 3 even. Thus, the smallest multiple of 300 that is a perfect square is:300×3=900300 \times 3 = 900300×3=900
3. Find the smallest number by which 1,458 should be multiplied to get a perfect square.
Now, let’s factorize 1,458:1458=2×33×71458 = 2 \times 3^3 \times 71458=2×33×7
For this number to be a perfect square, we need to make the exponents of all prime factors even. Currently, the exponents of 2, 3, and 7 are all odd. Therefore, we need to multiply by 2×3×7=422 \times 3 \times 7 = 422×3×7=42 to make all exponents even. Thus, the smallest number by which 1,458 should be multiplied is 42.
After multiplying:1458×42=612361458 \times 42 = 612361458×42=61236
Let’s check the square root of 61236:61236=246\sqrt{61236} = 24661236=246
So, the perfect square is 61236, and its square root is 246.
Final Answers:
- 300 is not a perfect square.
- The smallest multiple of 300 whose square root is an integer is 900 (since 900=30\sqrt{900} = 30900=30).
- The smallest number by which 1,458 should be multiplied to get a perfect square is 42.
- The square root of the perfect square 61236 is 246.
