if A=5^3*7^3*3^4, find the smallest number A must be multiplied to get a perfect square
you need all even exponents for a perfect square.
5^3 7^3 3^4 * 5*7 = 5^4 7^4 3^4
what is 7*5? Think about that. That is the smallest whole number.
But the smallest number you can multiply A and get a perfect square is
= 1/(5^3*7^3*3^2)
multiplying it then will leave you with A'=9, a perfect square. i am not certain your teacher was thinking of that, but if one asks a question, expect the answer.
well, technically, there is no smallest number. You could also multiply any perfect-square fraction with a larger denominator. That will give you an even smaller result. Not an integer, but a fraction as small as you like.
just sayin' ...
To find the smallest number A must be multiplied by to get a perfect square, we need to consider its prime factorization.
The prime factorization of A is: 5^3 * 7^3 * 3^4.
In order for A to be a perfect square, each exponent of its prime factors must be an even number. If the exponent is odd, multiplying A by a number would result in that exponent becoming even.
Let's analyze each prime factor's exponent:
1. The exponent of 5 is 3. To make it even, we need to multiply A by 5. So, we multiply A by 5.
A = 5 * (5^3 * 7^3 * 3^4) = 5^4 * 7^3 * 3^4.
Now, the exponents of both 5 and 3 are even because they are multiples of 2.
2. The exponent of 7 is already even, so we don't need to change anything for this prime factor.
Therefore, the smallest number A must be multiplied by to get a perfect square is 5^4.
The final expression becomes: A * 5^4 = 5^4 * 5^4 * 7^3 * 3^4, which is a perfect square.