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Why does 25 to the negative power of 1/2 equal 1/5?

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Lia Tan answered

Before we start to tackle the problem head on, let's break it down into two parts. First involving negative powers, and second involving powers as fractions.

So let's say we have the number x and we raise it to the power of -y. X^(-y) is then equal to the reciprocal of x raised to just y without the negative. This means that x^(-y) = (1/x)^(y). In your case, 25^(-1/2) = (1/25)^(1/2).

If you raise a number to another number that is a fraction, let's say 1/z, then this is equivalent into taking the zth root of that number. For instance, if you have 16 raised to the 1/4 or 16^(1/4), you are taking the fourth root of 16 (which is 2 since 2^4 = 16). Now that we simplified the problem to (1/25)^(1/2), we know that we are trying to find the second root or the square root of 1/25. The square root of 1/25 is 1/5.

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