## Surds and Indices Questions

Quadratic equations and inequalities, variation equations, function notation, systems of equations, etc.
firedude1337
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### Surds and Indices Questions

Hey guys, I'm writing here because I need some help with my homework. The questions I'm stuck on are:

$\left(\, \dfrac{ x^{\frac{7}{4}}\, -\, x^{\frac{3}{4}}\, +\, x\, \cdot\, x^{\frac{7}{4}} }{x^{\frac{1}{4}}}\, \right)^2$

and

$\left[\, \dfrac{y^{\frac{1}{2}}}{x^{\frac{3}{4}}}\, -\, \dfrac{x^{\frac{5}{4}}}{y^{\frac{3}{2}}}\, \right]^4$

I have to simplify them, but I have no idea how. Can someone please help and explain quite thoroughly? Thanks in advance!

maggiemagnet
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### Re: Surds and Indices Questions

I have to simplify them...

$\left(\, \dfrac{ x^{\frac{7}{4}}\, -\, x^{\frac{3}{4}}\, +\, x\, \cdot\, x^{\frac{7}{4}} }{x^{\frac{1}{4}}}\, \right)^2$
Where are you getting stuck? If you're needing to learn how to work with fractional powers, try here. They show how to simplify with exponents here. One of the things they say there is that there's usually more than 1 way to simplify these things. I think I'd start inside, like this:

$\dfrac{ x^{\frac{7}{4}}\, -\, x^{\frac{3}{4}}\, +\, x\, \cdot\, x^{\frac{7}{4}} }{x^{\frac{1}{4}}}\, =\, \dfrac{ x^{\frac{7}{4}}\, -\, x^{\frac{3}{4}}\, +\, x^{\frac{11}{4}} }{x^{\frac{1}{4}}}\,$

$=\, \left(\, x^{\frac{7}{4}}\, -\, x^{\frac{3}{4}}\, +\, x^{\frac{11}{4}}\, \right)\, \left(\, x^{-\frac{1}{4}}\, \right)$

Take the power through and simplify the fractions. Then square the result. I think it will be nasty, but I think that's probably the best you can do.
$\left[\, \dfrac{y^{\frac{1}{2}}}{x^{\frac{3}{4}}}\, -\, \dfrac{x^{\frac{5}{4}}}{y^{\frac{3}{2}}}\, \right]^4$
For this one, I'd start inside again.

$\dfrac{y^{\frac{1}{2}}}{x^{\frac{3}{4}}}\, -\, \dfrac{x^{\frac{5}{4}}}{y^{\frac{3}{2}}}\, =\, \dfrac{y^{\frac{1}{2}+\frac{3}{2}}\, -\, x^{\frac{5}{4}+\frac{3}{4}}}{x^{\frac{3}{4}}\, y^{\frac{3}{2}}}\, =\, \dfrac{y^2\,-\, x^2}{x^{\frac{3}{4}}\, y^{\frac{3}{2}}}$

Then you take the 4th power through. It'll be nasty this time, too.

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