# sqr(A+B*sqr(C))

Lately, I have been doing a lot of math involving square roots of numbers added to square roots, in the form of $$\sqrt{A+B\sqrt{C}}$$, this is called a “nested radical.” Normally, you would not be able to simplify any further, unless there was a common factor of both that could be removed, or if both items had the same number under the radical (eg. $$\sqrt{4\sqrt7+12\sqrt7}=\sqrt{16\sqrt7}=4\sqrt[4]{7}$$ ).

I can take a number like $$(1+\sqrt5)^2$$ and work out it is equal to $$6 + 2\sqrt5$$. Unfortunately, it is very difficult to recognize that $$1+\sqrt5$$ is the square root (or even a factor) of that. Even harder, trying to find an equation such as $$\sqrt{469+144\sqrt5}$$.

Recently, I found a method of breaking down the equation into its square root equation.

With an equation in the form of $$\sqrt{A+B\sqrt{C}}$$, where C contains no perfect square factors, taking the formula $$D^2 = A^2 – (B^2)*C$$, if D is a rational number, you will be able to factor the equation, otherwise it is simplified as much as is possible.

Then you will be able to get the formula: $$\sqrt{A+B\sqrt{C}} = \pm \left(\sqrt{\frac{A-D}{2}} + sign(A*B)*\sqrt{\frac{A+D}{2}}\right)$$, where sign(A*B) is the positive or negative value of A*B, it will always give you 0, +1, or -1.

So to find $$\sqrt{469+144\sqrt5}$$, we find $$469^2 – 144^2 * 5 = 116281$$. Since 116281 = 341*341, we can plug D=341 into the formula to get:

\begin{align} \sqrt{\frac{469-341}{2}} + 1*\sqrt{\frac{469+341}{2}}&=\sqrt{\frac{128}{2}} + \sqrt{\frac{810}{2}}\\ &=\sqrt{64} + \sqrt{405} \\ &= 8 + \sqrt{81*5} \\ &= 8+9\sqrt{5}\end{align}

So, $$\sqrt{469+144\sqrt5} = (8+9\sqrt{5}) \text{ and } (-8-9\sqrt{5})$$.

To find $$\sqrt{53-12\sqrt{11}}$$, we find $$53^2 – 12^2 * 11 = 1225$$. Since 1225 = 35*35, we can plug D=35 into the formula to get:

\begin{align*} \sqrt{\frac{53-35}{2}} + -1*\sqrt{\frac{53+35}{2}} &=\sqrt{\frac{18}{2}} – \sqrt{\frac{88}{2}}\\ &=\sqrt{9} – \sqrt{44} \\ &= 3 – \sqrt{4*11} \\ &= 3-2\sqrt{11} \end{align*}

So, $$\sqrt{53-12\sqrt11} = (3-2\sqrt{11}) \text{ and } (-3+2\sqrt{11})$$.

If A and B have the same sign, the answer will have both terms the same sign ({+,+} and {-,-} are equally valid). If A and B have different signs, the terms in the answer will have different signs ({+,-} and {-,+}). Often you will need to determine the complete answer that equates to positive, after all, it is hard to have a real object with a negative length.

An interesting correlation: $$(X-Y)^2 = (Y-X)^2$$, but remember $$\sqrt{X-Y} \not= \sqrt{Y-X}$$. Double check your signs by re-squaring your answer.

Note: This still works if D is not an integer, but is rational:

\begin{align*} \sqrt{8.41+2.64\sqrt5} &=\sqrt{\frac{8.41-5.99}{2}} + \sqrt{\frac{8.41+5.99}{2}} &D=\sqrt{8.41^2 + 2.64^2\cdot5}=5.99\\ &=\sqrt{\frac{2.42}{2}} + \sqrt{\frac{14.4}{2}}\\ &=\sqrt{\frac{121}{100}} + \sqrt{\frac{720}{100}}\\ &=\frac{11}{10} + \frac{12\sqrt5}{10}\\ &=1.1 + 1.2\sqrt5\\ \end{align*}

Which is correct. $$\left[1.1+1.2\sqrt5\right]^2=8.41+2.64\sqrt5$$