# What is x to the power of a decimal/fraction, e.g. 12^3.7

• What is x to the power of a decimal/fraction, e.g. 12^3.7

• @fabulouscamel, I think this can be solved if we understand fractional exponents. Suppose you have x^{1/n}. By the laws of exponents, if we raise that to the nth power, then it will give us x^1 which is x. So, basically x^{1/n} is the number whose nth power is x, like the square root is the number whose 2nd power is the original number. I think (don't quote me on this) that a positive number to a fractional exponent is always positive, so it's never negative. In your case, 12^3.7 can be written as 12^{37/10}, which is basically the number that taken to the 10th power is 12^37. You can try to put this in the calculator if you want; it's normally quite ugly and its decimals keep on running without pattern; we call that an irrational number.

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• @fabulouscamel

@energizedpanda basically covered it! Essentially, to find something like 12^(3.7), you could first turn the exponent into a fraction in simplest form-- i.e. 12^(37/10), and then we know that x^(a/b) = x^(a * 1/b) = (x^a)^(1/b) (this is an important exponent law), so 12^(37/10) = (12^37)^(1/10). Then, you would first calculate the giant number that is 12^(37), and take the 10th root of that. (Not so great to do by hand)

Hope that answered your question! 