Given that 𝑎 is equal to the square root of two and 𝑏 is equal to the square root of six, find the value of 𝑎 squared over 𝑏 squared.
In this problem, we need to work out what the value of 𝑎 squared over 𝑏 squared is. And, we need to use the values that we’re given for 𝑎 and 𝑏 to help us find it. 𝑎 squared over 𝑏 squared is written just like a fraction. So, we would expect that perhaps the value we’re looking for is going to be a fraction too. Let’s see. We’re told that the value of 𝑎 is the square root of two, and the value of 𝑏 is the square root of six. Because we know these values, we can substitute them for the letters 𝑎 and 𝑏 in the fraction.
And so, we can replace the 𝑎 in 𝑎 squared with the square root of two because 𝑎 is the same as the square root of two. And then, we can replace the 𝑏 in 𝑏 squared with the square root of six because we’re told that 𝑏 equals the square root of six. Unfortunately, instead of making the fraction looks simpler, we seem to have made it look a lot more complicated. The square root of two squared over the square root of six squared. What happens when you square a square root?
To help us understand this, let’s pick an easy number to work with. Let’s pick four times four. We know that four multiplied by four gives us a square number, which is 16. Now if we wrote the square root of 16, we’d expect the answer to be four because four times four equals 16. Let’s write the number four underneath here to remind ourselves that the square root of 16 is four. And if we go back to our square root and put brackets around it and then square it, this is the sort of thing that we’ve got going on in our fraction, the square root of 16 squared. We’ve already said that the square root of 16 is four. And if we square four, in other words we work out four times four, we get the answer 16.
It’s as if finding the square root of something and then squaring it cancel each other out. It’s also as if we can just erase all of the symbols around the number and just keep the number 16 itself. And so if we go back to our fraction, we can use the same idea to work out the answer. The numerator or the top number says the square root of two squared. We’ve already said that finding the square root of something and then squaring it cancels each other out. So, we’re left with just the number two. And, the same applies to the denominator or the bottom number. We’re just left with the number six. The value of 𝑎 squared over 𝑏 squared is equal to two-sixths.
Now, is there a way we could simplify this fraction? What if we divide the numerator and the denominator by two? Two divided by two equals one. And then, six divided by two gives us three. If 𝑎 is equal to the square root of two and 𝑏 is equal to the square root of six, then the value of 𝑎 squared over 𝑏 squared is two-sixths. Which we can simplify and write as one-third.