MaxWong

Using long division:

Quotient = -3x Remainder = 2x+2

Therefore \(\dfrac{-3x^3+5x+2}{x^2-1}=-3x+\dfrac{2x+2}{x^2-1}=-3x + \dfrac{2}{x-1}\)

Length of each side = sqrt(100)=10 in

Perimeter = 4 x 10 = 40 in

You made a careless mistake Guest #1, 

Guest : The length of the rectangle is 9 in. longer than the width.

Let x inch be the width.

\((x)(x+9)=52\\ x^2+9x-52=0\\ (x+13)(x-4)=0\\ x=-13(rejected)\;or\;x=4\)

The width of the rectangle is 4 in.

x^2 + 12^2 = 14^2

x^2 = 196 - 144 = 52

x = sqrt(52) = 2 sqrt(13)m

\(y = 5x^2 - 6x + 4\\ y' = 5\times 2 \times x^{2-1}-6\times 1 \times x^{1-1}+4\times 0 \times x^{0-1}=10x - 6\)

\(y = \dfrac{8}{x^7}\\ y'=8 \dfrac{d}{dx}x^{-7}=(8)(-7)(x^{-7-1})=-56x^{-8}=-\dfrac{56}{x^8}\)

Same as asinus's answer.

\(\sqrt[3]{117649}\times \sqrt[3]{4096}\\ = 49 \times 16\text{<--- according to mental maths}\\ = 784\)

You can learn the mental maths method below.....

Yeah! Of course.

\(\sqrt{2}^{\sqrt2}\text{<--- this looks like fun}\)

= \(2^{\frac{1}{\sqrt2}}\)

\(\sqrt2 ^ {\sqrt 2^{\sqrt2}}\text{ <--- this looks like more fun}\\ = 2^{\frac{1}{\sqrt2}\times \sqrt2}\\ = 2\)

Tips:Anything times 1 is itself.