Does anyone know how to do radical integers on a calculator or how to program one to do them

All scientific and graphing calculators have the ability to find square roots. Usually, the key involved has a square-root symbol: √. The particulars can be found in the manual for your calculator. (Sometimes, a shift operation is required to access that key.)

If you only have a 4-function calculator, you can still get close, but it takes several steps. The basic method is to divide the original number by a guess at its square root, then average that result with the guess to make a new guess. Continue until you have the accuracy you desire.

For example, √17 ≈ 4. Dividing 17 by 4 gives 4.25, and the average of that with 4 is 4.125. Dividing 17 by 4.125 gives 4.121212... and the average of that with 4.125 is 4.12310606... This is good to 5 significant digits. (4.1231).

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If you're concerned with roots of index other than 2, you can use a similar method to the above, but it gets more complicated for higher roots. For example, the next guess for a cube root of n is (2/3)x + (1/3)n/x², where x is the present guess.

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For calculators with a yˣ key, the value of x can be the reciprocal of the root index. For a 5th root, you would use x=1/5. That is ...

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You can always use logarithms to find roots. The log of the root is the log of the original number divided by the root index. That is

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