Question

$$\left. \begin{array} { l } { \frac { 1 } { \sqrt { 4 + \sqrt { 5 } } } + \frac { 1 } { \sqrt { 5 + \sqrt { 6 } } } + \frac { 1 } { \sqrt { 6 } + \sqrt { 7 } } + \frac { 1 } { \sqrt { 7 + \sqrt { 8 } } } + \frac { 1 } { \sqrt { 8 } + \sqrt { 9 } } \right.$$

Answer

$$o=(-1-sqrt(5)-sqrt(6)-sqrt(7)-2*sqrt(2))/(Pr*e*v*t^2*h*a)$$

Solution


Since \(2\times 2=4\), the square root of \(4\) is \(2\).
\[Provethat\times 1\times 2+\sqrt{5}+1\times \sqrt{5}+\sqrt{6}+1\times \sqrt{6}+\sqrt{7}+1\times \sqrt{7}+\sqrt{8}+1\times \sqrt{8}+\sqrt{9}=1\]
Simplify  \(\sqrt{8}\)  to  \(2\sqrt{2}\).
\[Provethat\times 1\times 2+\sqrt{5}+1\times \sqrt{5}+\sqrt{6}+1\times \sqrt{6}+\sqrt{7}+1\times \sqrt{7}+2\sqrt{2}+1\times 2\sqrt{2}+\sqrt{9}=1\]
Since \(3\times 3=9\), the square root of \(9\) is \(3\).
\[Provethat\times 1\times 2+\sqrt{5}+1\times \sqrt{5}+\sqrt{6}+1\times \sqrt{6}+\sqrt{7}+1\times \sqrt{7}+2\sqrt{2}+1\times 2\sqrt{2}+3=1\]
Regroup terms.
\[2ovtthaPre+\sqrt{5}+1\times \sqrt{5}+\sqrt{6}+1\times \sqrt{6}+\sqrt{7}+1\times \sqrt{7}+2\sqrt{2}+1\times 2\sqrt{2}+3=1\]
Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[2ov{t}^{2}haPre+\sqrt{5}+1\times \sqrt{5}+\sqrt{6}+1\times \sqrt{6}+\sqrt{7}+1\times \sqrt{7}+2\sqrt{2}+1\times 2\sqrt{2}+3=1\]
Regroup terms.
\[2Preov{t}^{2}ha+\sqrt{5}+1\times \sqrt{5}+\sqrt{6}+1\times \sqrt{6}+\sqrt{7}+1\times \sqrt{7}+2\sqrt{2}+1\times 2\sqrt{2}+3=1\]
Simplify  \(1\times \sqrt{5}\)  to  \(\sqrt{5}\).
\[2Preov{t}^{2}ha+\sqrt{5}+\sqrt{5}+\sqrt{6}+1\times \sqrt{6}+\sqrt{7}+1\times \sqrt{7}+2\sqrt{2}+1\times 2\sqrt{2}+3=1\]
Simplify  \(1\times \sqrt{6}\)  to  \(\sqrt{6}\).
\[2Preov{t}^{2}ha+\sqrt{5}+\sqrt{5}+\sqrt{6}+\sqrt{6}+\sqrt{7}+1\times \sqrt{7}+2\sqrt{2}+1\times 2\sqrt{2}+3=1\]
Simplify  \(1\times \sqrt{7}\)  to  \(\sqrt{7}\).
\[2Preov{t}^{2}ha+\sqrt{5}+\sqrt{5}+\sqrt{6}+\sqrt{6}+\sqrt{7}+\sqrt{7}+2\sqrt{2}+1\times 2\sqrt{2}+3=1\]
Simplify  \(1\times 2\sqrt{2}\)  to  \(2\sqrt{2}\).
\[2Preov{t}^{2}ha+\sqrt{5}+\sqrt{5}+\sqrt{6}+\sqrt{6}+\sqrt{7}+\sqrt{7}+2\sqrt{2}+2\sqrt{2}+3=1\]
Simplify  \(2Preov{t}^{2}ha+\sqrt{5}+\sqrt{5}+\sqrt{6}+\sqrt{6}+\sqrt{7}+\sqrt{7}+2\sqrt{2}+2\sqrt{2}+3\)  to  \(2Preov{t}^{2}ha+2\sqrt{5}+2\sqrt{6}+2\sqrt{7}+4\sqrt{2}+3\).
\[2Preov{t}^{2}ha+2\sqrt{5}+2\sqrt{6}+2\sqrt{7}+4\sqrt{2}+3=1\]
Subtract \(2\sqrt{5}\) from both sides.
\[2Preov{t}^{2}ha+2\sqrt{6}+2\sqrt{7}+4\sqrt{2}+3=1-2\sqrt{5}\]
Subtract \(2\sqrt{6}\) from both sides.
\[2Preov{t}^{2}ha+2\sqrt{7}+4\sqrt{2}+3=1-2\sqrt{5}-2\sqrt{6}\]
Subtract \(2\sqrt{7}\) from both sides.
\[2Preov{t}^{2}ha+4\sqrt{2}+3=1-2\sqrt{5}-2\sqrt{6}-2\sqrt{7}\]
Subtract \(4\sqrt{2}\) from both sides.
\[2Preov{t}^{2}ha+3=1-2\sqrt{5}-2\sqrt{6}-2\sqrt{7}-4\sqrt{2}\]
Subtract \(3\) from both sides.
\[2Preov{t}^{2}ha=1-2\sqrt{5}-2\sqrt{6}-2\sqrt{7}-4\sqrt{2}-3\]
Simplify  \(1-2\sqrt{5}-2\sqrt{6}-2\sqrt{7}-4\sqrt{2}-3\)  to  \(-2-2\sqrt{5}-2\sqrt{6}-2\sqrt{7}-4\sqrt{2}\).
\[2Preov{t}^{2}ha=-2-2\sqrt{5}-2\sqrt{6}-2\sqrt{7}-4\sqrt{2}\]
Divide both sides by \(2\).
\[Preov{t}^{2}ha=\frac{-2-2\sqrt{5}-2\sqrt{6}-2\sqrt{7}-4\sqrt{2}}{2}\]
Factor out the common term \(2\).
\[Preov{t}^{2}ha=\frac{-2(1+\sqrt{5}+\sqrt{6}+\sqrt{7}+2\sqrt{2})}{2}\]
Move the negative sign to the left.
\[Preov{t}^{2}ha=-\frac{2(1+\sqrt{5}+\sqrt{6}+\sqrt{7}+2\sqrt{2})}{2}\]
Cancel \(2\).
\[Preov{t}^{2}ha=-(1+\sqrt{5}+\sqrt{6}+\sqrt{7}+2\sqrt{2})\]
Remove parentheses.
\[Preov{t}^{2}ha=-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}\]
Divide both sides by \(Pr\).
\[eov{t}^{2}ha=\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Pr}\]
Divide both sides by \(e\).
\[ov{t}^{2}ha=\frac{\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Pr}}{e}\]
Simplify  \(\frac{\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Pr}}{e}\)  to  \(\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Pre}\).
\[ov{t}^{2}ha=\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Pre}\]
Divide both sides by \(v\).
\[o{t}^{2}ha=\frac{\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Pre}}{v}\]
Simplify  \(\frac{\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Pre}}{v}\)  to  \(\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Prev}\).
\[o{t}^{2}ha=\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Prev}\]
Divide both sides by \({t}^{2}\).
\[oha=\frac{\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Prev}}{{t}^{2}}\]
Simplify  \(\frac{\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Prev}}{{t}^{2}}\)  to  \(\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Prev{t}^{2}}\).
\[oha=\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Prev{t}^{2}}\]
Divide both sides by \(h\).
\[oa=\frac{\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Prev{t}^{2}}}{h}\]
Simplify  \(\frac{\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Prev{t}^{2}}}{h}\)  to  \(\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Prev{t}^{2}h}\).
\[oa=\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Prev{t}^{2}h}\]
Divide both sides by \(a\).
\[o=\frac{\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Prev{t}^{2}h}}{a}\]
Simplify  \(\frac{\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Prev{t}^{2}h}}{a}\)  to  \(\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Prev{t}^{2}ha}\).
\[o=\frac{-1-\sqrt{5}-\sqrt{6}-\sqrt{7}-2\sqrt{2}}{Prev{t}^{2}ha}\]