Solution to Nested Radical Expressions

This is the exciting part of quadratic equations. It is fun to see dancing radicals like this \sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3\ldots}}}}. This is an unending chain of radical expression and can be surprisingly simplified using quadratic equations.

Worked Problem 1:

\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\ldots}}}} can be simplified in the form of \displaystyle\frac{p+\sqrt{q}}{r}. Find the value of p+q+r

 

Solution:

Let x=\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\ldots}}}}

By squaring both sides;

x^2=(\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\ldots}}}})^2

x^2=3+\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\ldots}}}}

Since x=\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\ldots}}}}, by substitution

x^2=3+x or x^2-x-3=0

Since we are looking for x we can use quadratic formula to solve for it.

x_1x_2=\displaystyle\frac{-b\pm\sqrt{b^2-4ac}}{4a}

x_1x_2=\displaystyle\frac{-(-1)\pm\sqrt{(-1)^2-4(1)(-3)}}{4(1)}

x_1x_2=\displaystyle\frac{1\pm\sqrt{13}}{2}

Only keep the positive x.

x=\displaystyle\frac{1+\sqrt{13}}{2}

Therefore, p+q+r=1+13+2=16

 

Worked Problem 2:

Simplify the radical \sqrt{5\sqrt{5\sqrt{5\sqrt{5\ldots}}}}.

 

Solution:

Let x=\sqrt{5\sqrt{5\sqrt{5\sqrt{5\ldots}}}}

By squaring both sides

x^2=(\sqrt{5\sqrt{5\sqrt{5\sqrt{5\ldots}}}})^2

x^2=5\sqrt{5\sqrt{5\sqrt{5\sqrt{5\ldots}}}}

Since x=\sqrt{5\sqrt{5\sqrt{5\sqrt{5\ldots}}}} by substitution

x^2=5x or x^2-5x=0

By factoring

x(x-5)=0

x=0 and x=5

 

Worked Problem 3:

Find x if x in. What is the value of x?

Solution:

Since  x inby substitution

x^2=2

x=\sqrt{2}

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