Solve $\sqrt{x+16}+\sqrt[4]{x+16}=12$.

Solve $\sqrt{x+16}+\sqrt[4]{x+16}=12$.

For some students, they would think to square the given equation three times to get rid of the fourth root and square root terms:

$\sqrt{x+16}+\sqrt[4]{x+16}=12$

$(\sqrt[4]{x+16})^2=(12-\sqrt{x+16})^2$

$\sqrt{x+16}=144-24\sqrt{x+16}+x+16$

Mathematical Problem Solving Skill

In this blog post, I will show one really insightful method given by one very intelligent U.K. mathematician on how he used his own elegant way to prove that $\tan^2 20^{\circ}+\tan^2 40^{\circ}+\tan^2 80^{\circ}=33$.

He first noticed that

$\tan 3(20^\circ)=\tan 60^\circ=\sqrt{3}$

$\tan 3(40^\circ)=\tan 120^\circ=-\sqrt{3}$

$\tan 3(80^\circ)=\tan 240^\circ=\sqrt{3}$

Floor Function Problem...

Solve the following equation:

$\displaystyle \left\lfloor x+\frac{7}{3} \right\rfloor^2-\left\lfloor x-\frac{9}{4} \right\rfloor=16$

Note: $\displaystyle \lfloor x \rfloor$ denotes the largest integer not greater than $x$. This function, referred to as the floor function, is also called the greatest integer function, and its value at $x$ is called the integral part or integer part of $x$.

Evaluate $abcd$

Let $a, b, c, d$ be real numbers such that [MATH]a=\sqrt{4-\sqrt{5-a}}[/MATH], [MATH]b=\sqrt{4+\sqrt{5-b}}[/MATH], [MATH]c=\sqrt{4-\sqrt{5+c}}[/MATH] and [MATH]d=\sqrt{4+\sqrt{5+d}}[/MATH]. Calculate $abcd$.

This is one very intriguing problem because I am sure your instinct will tell you, No, please don't multiply all the given four expression out! That is a surest way to get into a big mess if you do that, and there got to be an easy way out for this problem.

(2) Evaluate The Sum Of 1/xy+z-1+1/yz+x-1+1/xz+y-1

Regarding the problem to evaluate:

[MATH]\frac{1}{xy+z-1}+\frac{1}{yz+x-1}+\frac{1}{xz+y-1}[/MATH]

given:

[MATH]\begin{cases}x+y+z=2 \\[3pt] x^2+y^2+z^2=3 \\[3pt] xyz=4 \\ \end{cases}[/MATH]

you might be wondering if the proposed solution is the only way to stab at the problem, or more specifically, if you solved it differently, but in a more tedious or long way, can you be proud of it?