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

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

# 3 Answers

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.

25^{3/2} can be easily rewritten by taking the square root first. I'd rather not deal with cubing 25.

25^{3/2 }= 5^{3} = 125.

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

**(b ^{m})^{n} = b^{(m*n)}**

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

**[(25 ^{3})^{(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

**[(25 ^{3})^{(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

## Comments

Indeed.

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