Algebra Word Problem Solutions

Roots of a quadratic equation.
Given: ${\displaystyle a{\left(f\left(x\right)\right)}^2+bf\left(x\right)+c=0.}$

Rewrite the following exponential expressions as equivalent radical expressions. If the number is rational, write it without radicals or exponents.
${\displaystyle {27}^{-\frac{2}{3}}}$

Rewrite the following exponential expressions as equivalent radical expressions. If the number is rational, write itwithout radicals or exponents.
${\displaystyle 4^{-\frac{3}{2}}}$

Rationalization of ${\displaystyle \frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}+\sqrt{5}}}$
My approach:
I tried to rationalize the denominator by multiplying it by ${\displaystyle \frac{\sqrt{2}-\sqrt{3}-\sqrt{5}}{\sqrt{2}-\sqrt{3}-\sqrt{5}}}$. And got the result to be (after a long calculation):
${\displaystyle \frac{\sqrt{24}+\sqrt{40}-\sqrt{16}}{\sqrt{12}+\sqrt{5}}}$
which is totally not in accordance with the answer, ${\displaystyle \sqrt{2}+\sqrt{3}-\sqrt{5}}$
Can someone please explain this/give hints to me.

Rationalizing fractions with multiple radicals
I am having trouble rationalizing ${\displaystyle \frac{2}{\sqrt{2-\sqrt{2}}}}$
I have tried multiplying the fraction by ${\displaystyle \sqrt{2+\sqrt{2}}}$ and got ${\displaystyle \frac{2\sqrt{2+\sqrt{2}}}{\sqrt{2}}}$ am not sure if that is correct or not but I then multiplied by ${\displaystyle \sqrt{2}}$ but got stuck...
Assuming that I did the problem correctly so far, how would I multiply ${\displaystyle 2\sqrt{2+\sqrt{2}}}$ with ${\displaystyle \sqrt{2}}$

Recursive equations for quadratic diophantine equations.
How do I determine the recursive equations for deriving integer solutions to quadratic Diophantine equations. Say I have a Diophantine equation. e.g.
${\displaystyle 5x^2-3x+3y^2+2y=0}$
How do I determine a recursive equation so that with one integer solution I can determine the other integer solutions.

Range of an expression related to coefficients of a quadratic equation
Let ${\displaystyle a{x}^2+bx+c=0}$ be a quadratic equation such that both the roots lie in [0,1]. What is the range of the expression ${\displaystyle \frac{(a-b)(2a-b)}{a(a-b+c)}}$?

Range of p satisfying quadratic inequality
What is the set of values of p for which ${\displaystyle p(x^2+2)<2x^2+6x+1}$, for all real values of x?

Question about the square root of a non-negative quadratic function.
For which values of x does this apply
${\displaystyle \sqrt{x^2+2x+1}=x+1}$

${\displaystyle y=4.1x+2}$ and ${\displaystyle 2y=4.1x+2}$

${\displaystyle x^2+2ax+b=0}$ where ${\displaystyle x_1}$ and ${\displaystyle x_2}$ are real solutions.
For which value of b function ${\displaystyle f\left(b\right)=|x_1-x_2|}$ reaches maximum if ${\displaystyle |x_1-x_2|=2m}$

Prove ${\displaystyle {10}^{\frac{P\left(n\right)}{2}}+1}$ is divisible by n

Proof the logaritmic identity
Please help me proving the basic logarithmic identity
${\displaystyle {\log }_{3}12=1+{\log }_{5}4\cdot {\log }_{6}5\cdot {\log }_{3}6}$

The equation ${\displaystyle {\sin }^{4}x-(k+2){\sin }^{2}x-(k+3)=0}$ possesses a solution if:
A) ${\displaystyle k\succ 3}$
B) ${\displaystyle k<-2}$
C) ${\displaystyle -3\le k\le -2}$
D) k is any (+ve) value

Suppose your weight is 53.81 kilograms. A scale at a health clinic gives your weight as 54.4 kilograms. A digital scale at the gym that gives readings to the nearest 0.01 kilogram gives your weight as 54.19 kilograms. Which measurement is more​ precise? Which is more​ accurate?
Which measurement is more​ precise? Choose the correct answer below.
A. The digital scale at the gym is more precise.
B. The health clinic scale is more precise.
Which measurement is more​ accurate? Choose the correct answer below.
A. The digital scale at the gym is more accurate.
B. The health clinic scale is more accurate.