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# Bina-ring

Consider $(1 + \sqrt2)^n$. This will be of the form $A + B\sqrt2$ where $A$ and $B$ are integers. Decide which entries in the table below are possibe and which are not.

What happens for $(a + \sqrt p)^n$ for other values of $p$?

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Age 16 to 18

Challenge Level

Consider $(1 + \sqrt2)^n$. This will be of the form $A + B\sqrt2$ where $A$ and $B$ are integers. Decide which entries in the table below are possibe and which are not.

A even | A odd | |

B even | ||

B odd |

What happens for $(a + \sqrt p)^n$ for other values of $p$?

Find the value of sqrt(2+sqrt3)-sqrt(2-sqrt3)and then of cuberoot(2+sqrt5)+cuberoot(2-sqrt5).

Find the smallest numbers a, b, and c such that: a^2 = 2b^3 = 3c^5 What can you say about other solutions to this problem?