Fractional Indices questions

How will I work out these questions .

1. 5x (square root of 5) can be written in the form 5^n
Calculate the value of n .

2. a= 2^x b = 2^y
Express in terms of a and b
A) 2^x+y
B) 2^2x
C) 2(^x+2y)

x

Just to rephrase what @luckybillion said (so I don't steal the credit lol),

Part 1

Understand this rule that $\displaystyle (\sqrt[a]{k})^{b} \equiv k^{\frac{b}{a}}$

Easy way to remember this is to remember that $\displaystyle \sqrt[a]{k} = k^{\frac{1}{a}}$ and then the outside power so $\displaystyle (k^{\frac{1}{a}})^{b}$ and this brings you back to $\displaystyle k^{\frac{b}{a}}$ when you expand the brackets!

Part 2

Given that $a = 2^{x}$ and $b = 2^{y}$

a) Firstly, you should've put the x+y into brackets because I interpreted 2^x+y as (2^x)+y and not 2^(x+y) which are both different.

Anyway, the rule you use to combine the powers given the bases are the same is multiplying both $a$ and $b$

i.e. $a \times b = 2^{x} \times 2^{y} = 2^{x+y}$

So a general rule is $n^{a} \times n^{b} = n^{a+b}$

b) The general rule here is that $(n^{a})^{k} = n^{a \times k} = n^{ak}$

So in your case, $2^{2x} = (2^{x})^{2} = a^{2}$

c) So you need to obtain $2^{x+2y}$

Use the previous two rules you used.

We know that $2^x = a$ and $2^{2y} = b^{2}$

Therefore combining the powers together (i.e. multiply) then you get $2^{x} \times 2^{2y} = a \times b^{2} = ab^{2} = 2^{x+2y}$

You'll need to remember some basic rules. The square root is the same as to the power of a half. So for 1, n is 1/2.
2a) the answer is a*b. because you add the powers when you multiply the same bases together
2b) (2^x)^2 . because you would multipy 2 and x together when the whole thing is raised by another power.
2c) combining both part a and b knowledge.
it would be (2^y)^2 * 2^x