A Root of a Root

Algebra Level 2

The number $\sqrt{27 + \sqrt{200} }$ can be simplified to the form $a + \sqrt{b}$ , where $a$ and $b$ are positive integers. Find the product $ab$ .

Details and assumptions

This problem first appeared in Christoff Rudolff's book Coss , which is perhaps the first German algebra book. It was written in 1525. Rudoff was the first to use our notation for the square root.

The answer is 10.

1 solution

Arron Kau Staff
May 13, 2014

If we set $\sqrt{27 + \sqrt{200} }$ equal to $a + \sqrt{b}$ and square both sides, we get the equation $27 + \sqrt{200} = (a + \sqrt{b})^2$ . Simplifying, expanding, and grouping like terms gives us $27 + 10\sqrt{2} = a^2 + b + 2a \sqrt{b}$ . It is then easy to see from inspection that $b = 2$ , $a = 5$ is a solution, so that $\sqrt{27 + \sqrt{200} } = 5 + \sqrt{2}$ . Therefore the product $ab$ is 10.

Oh, so this is the first of its type! A really famous problem in my school!

Swapnil Das - 5 years, 11 months ago

A Root of a Root

The answer is 10.

1 solution

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