I won't lie, I have the following questions in my maths homework but I have no idea how to solve it and I wondered if you could help me here! My teacher has taught us, what seems to be the basics of surds now, but it no seems quite apparent that he hasn't taught us everything – still, he wants to test us seeing as though we are doing a maths GCSE is November; so I get the impression that with the knowledge we already have, it would be possible to work the question out eventually. But anyway, the question is as follows:
Given that $135 = 3^3 \cdot 5$, simply the expression – Give your answer in surd form
$$\frac{\sqrt{135}}{\sqrt{7}-\sqrt{12}}$$
Any help would be greatly appreciated, although I straight answer isn't really that helpful to anyone as I really need to know how you got the answer – so any explainations would be brilliant!
Thanks in advance
Everyone,
Thank you very much for all the help you have provided (or attempted to provide 🙂 )! I am so reluctant and somewhat afraid to announce it, but it turns out that there was an error in the question. When I went back to maths, my maths teacher admitted that even he struggled with the question because it was not the question it was supposed to be, and he said "the board that produce these example exam questions must have made a typo…", as it is supposed to be ${\sqrt{75}}$ not ${\sqrt{7}}$! Suddenly the question is considerably more easy to solve….
So, once again, much appreciation for everyone who has posted on this question, and apologies for wasting people's time!
Best Answer
Here's a similar problem worked out (assuming that "in surd form" means something like "as a sum of roots of square-free integers multiplied by rational numbers"):
$$ \begin{align} \frac{\sqrt{875}}{\sqrt{11}-\sqrt{45}} &=\frac{\sqrt{875}(\sqrt{11}+\sqrt{45})}{(\sqrt{11}-\sqrt{45})(\sqrt{11}+\sqrt{45})}\\ &=\frac{\sqrt{875}(\sqrt{11}+\sqrt{45})}{\sqrt{11}^2-\sqrt{45}^2}\\ &=\frac{\sqrt{875}(\sqrt{11}+\sqrt{45})}{11-45}\\ &=\frac{\sqrt{5^3\cdot7}(\sqrt{11}+\sqrt{3^2\cdot5})}{-34}\\ &=\frac{5\sqrt{5\cdot7}(\sqrt{11}+3\sqrt{5})}{-34}\\ &=\frac{5\sqrt{5\cdot7\cdot11}+15\sqrt{5\cdot5\cdot7}}{-34}\\ &=\frac{5\sqrt{5\cdot7\cdot11}+75\sqrt{7}}{-34}\\ &=-\frac5{34}\sqrt{385}-\frac{75}{34}\sqrt{7}\\ \end{align} $$