If 𝑥 is equal to square root five over two square root three and 𝑦 is equal to one over square root six, is it true that two 𝑥 squared plus 𝑦 squared equals one?
To decide if this is true or not, we simply need to plug in the values for 𝑥 and 𝑦 and find out if our equation turns out to be something that is true. So for two 𝑥 squared plus 𝑦 squared equals one, we need to replace 𝑥 with square root five over two square root three and replace 𝑦 with one over square root six.
So our first step is to take care of these exponents, the squares. An what we know is that the power of a quotient of two numbers is distributed over the numerator and the denominator. So essentially, we need to raise the numerator to that power and denominator to that power. So we’re taking the numerators and denominators to the second power.
So let’s bring down the two and now work inside the brackets. So on the numerator, we have the square root of five being squared, which is equal to five, because a square will get rid of a square root. Or we can think if we square square root five, we have square root 25. And the square root of 25 is indeed five. Now the denominator, we have to be careful. We need to square the two on the outside of the square root and square the square root of three and multiply these numbers together.
So two squared is four. And if we squared square root three, we have a square root of nine or just three. And then on the denominator, four times three is equal to 12. Now for the second fraction, one squared is one. And if we squared square root six, we get six because a square root of 36, because we will take square root six squared, which is the square root of 36, which is six. Now the two that’s in the very front we could rewrite as two over one. And this will make it easier to see that we can reduce, because two goes into itself once and goes into 12 six times.
So this first fraction becomes five times one, the numerator, which is five and one times six, the numerator, which is six. So now we need to take five-sixths and add one-sixth. Well when we add fractions they must have a common numerator, which they do. They’re both sixths. And now we add the number on the numerator. So five plus one is equal to six. Now six over six does reduce because six divided by six is one.
So do we end up for something that is true? Is one equal to one? Yes, it is. So to answer our question: yes, it’s true.