1. ## indices - help!

I have many questions that I need help with.

1. Express 27 (to the power of x) in the form 3
(to the power of ax) where a is an integer to be found.

1(ii). hence solve the equation 3 (to the power of x²) = 27 (to the power of x)

2. √90 in the form a√10 where a is an integer

2(ii). Express (5-2
√3) ² in the form b+c√3 where b and c are integers

3(i). Express 15/2-√2 in the form p+q√2

3(ii). Express √8+√2 / √8-√2 as a single fraction

I have many questions that I need help with.

1. Express 27 (to the power of x) in the form 3
(to the power of ax) where a is an integer to be found.

1(ii). hence solve the equation 3 (to the power of x²) = 27 (to the power of x)

I'll try to interpret what you want, but I'm afraid something's lost in the translation. You may want to try and restate this one.

$\displaystyle 27^x=3^{ax}$

$\displaystyle 3^{3x}=3^{ax}$

$\displaystyle 3x=ax$

$\displaystyle 3=a$
2. √90 in the form a√10 where a is an integer

$\displaystyle \sqrt{90}=\sqrt{9\cdot10}=3\sqrt{10}$

2(ii). Express (5-2 √3) ² in the form b+c√3 where b and c are integers

$\displaystyle (5-2\sqrt{3})^2=25-20\sqrt{3}+12=37-20\sqrt{3}$

$\displaystyle \frac{15}{2-\sqrt{2}}\cdot\frac{2+\sqrt{2}}{2+\sqrt{2}}=\frac{ 30+15\sqrt{2}}{4-2}=15+\frac{15}{2}\sqrt{2}$
$\displaystyle \frac{\sqrt{8}+\sqrt{2}}{\sqrt{8}-\sqrt{2}}\cdot\frac{\sqrt{8}+\sqrt{2}}{\sqrt{8}+\s qrt{2}}=\frac{8+2\sqrt{16}+2}{8-4}=\frac{8+8+2}{4}=\frac{18}{4}=\frac{9}{2}$