# Evaluating expression with Integer part and Fraction part of a nested radical

Let $$A= \sqrt{93+28\sqrt{11}}$$ if $$B$$ is the integer part of $$A$$ and $$C$$ is the fraction part of $$C$$, what is the value of $$B+C^2$$

I tried manipulating it by setting

$$A=B+C$$

but I can't transform it into the expression, do I need approximate the integer part of A in order to solve this or is there another way?

• Presumably you mean $C$ is the fractional part of $A$ – Ross Millikan Mar 23 at 14:07
• What do you mean by fractional part? That number is irrational. – Allawonder Mar 23 at 14:10

Let's find intgers $$x,y$$ such that $$(x+y\sqrt{11})^2= 93+28\sqrt{11}$$

Then $$x^2+11y^2 =93 \;\;\;\wedge \;\;\; 2xy = 28$$

Since $$11y^2<99\implies y^2 <9 \implies |y|\leq 2$$ and $$xy=14$$. Playing with numbers we see that $$x=7$$ and $$y=2$$ works well, so $$\boxed{\sqrt{93+28\sqrt{11}} = 7+2\sqrt{11}}$$

Since $$13= 7+2\cdot 3<7+2\sqrt{11} <7+2\cdot \sqrt{49\over 4} = 14$$

we see $$B =13$$...

Hint:

$$(7+2\sqrt{11})^2=\cdots$$