Express five halves square root 252 in the form 𝑎 square root 𝑏 where 𝑎 and 𝑏 are both integers and 𝑏 takes the least possible value.
So here, we have our expression. And we want to change it where 𝑎, which is gonna be in the five-halves spot, and 𝑏, which is on the inside of the square root, are both integers. Integers of the nice numbers. They’re the negatives, zero, and the positive numbers. Negative two, negative one, zero, one, two, going more negative and more positive.
Now, the important piece is that 𝑏 needs to take the least possible value. So we need to simplify inside this square root. Well, 252, we can break down into its factors. 252 is the same as four times 63. 63 is nine times seven. The nice thing is nine is a perfect square and so is four. And then, seven is prime. So it doesn’t break down anymore.
So we can rewrite this as five-halves times the square root of four times nine times seven. And we can separate the pieces inside the square root. However, we know the square root of four is two and the square root of nine is three. And the square root of seven does not break down.
So, our tows can cancel. And we’re left with five times three which is 15. And then, we have square root seven. And seven is the least possible value that we could have for 𝑏. Because now that it’s prime, it won’t break down anymore. It can’t get any smaller. So 15 square root seven will be our final answer.