What number should b multiplied by 3610 to make it a perfect square. please give me a way to solve it?

Answer 1

Answer: b=10 We set up an equation that at first looks hard to solve because we have only one equation and two unkonwns, but it ends up being simple. The equations is: 3610b=r^2. We want r to be a natural number. So square root of 3610xr must also be a natural number. Now look at something simple, such as 6b=r^2, in other words what would we multiply 6 by to get a perfect square. We know the answer is 6 since 6x6=36. But why? Understanding why will make our problem very easy. If you have the product of perfect squares under the readical sign, then the square root if a natural number. Now note that 6 is 2x3 and neither is a perfect square. But 2^2x3^2 is36 which is. What I did was to make sure I has 2^2 and 3^2 so the square root would be a natural number. I kind of complete the square by putting in the missing numbers so that the exponent on each number in the prime factorization is square. So now to your question. 3610=2x5x19^2, so to make it a perfect square we need to mutliply by 2 first to make the 2 we have into 2^2 and then by 5 to make the into 5^2. In other words b must be 10 That means 3610 x10=36100 and square root of 36100 is190 so we are done.

In this case, when we clearly see that 36 is a perfect square , then we write √3610 = √(36 x 10)

and since we know that 10 x 10 is a perfect square, then

√(3610b) = √(3610 x 10) = √(36100) = √(36 x 100) = √36√10 = 6 x 10 = 60