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# What is 25 raised to the negative 3/2 power?

What is 25 raised to the negative 3/2 power?

Hi Martin,

An exponent of the form 1/n for a number n is taken to mean taking the nth root of the number. In this case, we're just working with n=2, which means a square root and should be familiar to you.

253/2 can be easily rewritten by taking the square root first. I'd rather not deal with cubing 25.

253/2 = 53 = 125.

Indeed.

One of the basic properties of exponents is

b(-1) = 1/b

Also, a base raised to an exponent that's less than one (e.g. 1/2) means the same as taking the square root:

b(1/2) = √b

Yet another exponent rule is the power rule

(bm)n = b(m*n)

So another way that 25(-3/2) could be expressed is

[(253)(1/2)](-1)

since multiplying the exponents based on the power rule yields the original expression. Therefore the problem could be stated as the inverse of the square root of 25 cubed

Putting it all together gives

[(253)(1/2)](-1)  =  [√(15625)]-1 = [125]-1 = 1/125 = 0.008

25^(-3/2)

= (5^2)^(-3/2)

= 5^(-3)

= 1/5^3

= 1/125