Given that 𝑎 over five is the multiplicative inverse of root five over 11, find the value of 𝑎.
Well, in this question, the first thing we need to look at is the multiplicative inverse. What is it? Well, what we know about multiplicative inverse is that the product of a number and its multiplicative inverse is one. So therefore, in our question, we know that 𝑎 over five multiplied by root five over 11 must be equal to one.
And then to help us solve this, we can use one of our operation rules for fractions. And this tells us that if we have 𝑎 over 𝑏 multiplied by 𝑐 over 𝑑, it’s equal to 𝑎𝑐 over 𝑏𝑑. So we multiply the numerators and the denominators. So we’re gonna get 𝑎 root five, cause that’s the numerators multiplied, over 55, because five multiplied by 11 is 55, is equal to one.
So therefore, the next step will be to multiply each side of the equation by 55. And when we do this, we get 𝑎 root five is equal to 55. And then, if we divide both sides of the equation now by root five, because we’re trying to find 𝑎, we get 𝑎 is equal to 55 over root five.
So now we’ve got to the stage where we found 𝑎. What we want to do is simplify it because we want to rationalize the denominator, cause we want to avoid having a surd as the denominator. And to do this, what we’re gonna do is multiply both the numerator and denominator by the surd or the root that’s on the denominator.
So we’re gonna have 55 over root five multiplied by root five over root five. So this is gonna give us 𝑎 is equal to 55 root five over five. And this is because we have a relationship with surds or roots. And that is that root 𝑎 multiplied by root 𝑎 will give us root 𝑎 squared. Well, root 𝑎 squared is just equal to 𝑎.
So in this case, if you have root five multiplied by root five, it gives us root 25. And root 25 is just five. So therefore, we can say that if we divide 55 root five by five, we’re gonna get 11 root five.
So therefore, given that 𝑎 over five is the multiplicative inverse of root five over 11, the value of 𝑎 is 11 root five.