Understanding Exponents: Solving the Power Question

[eddysd]

I know this is really simple, but it's been a while since I studied maths, but when you have something to the power 3/2, say x for example, would it be sqrt(x^3) or (sqrt(x))^3?

 

[HallsofIvy]

Either one- they are equal. If x= 25, say, then $x^3= 25^3= 15625$ so $\sqrt{x^3}= \sqrt{15625}= 125$ while $\sqrt{x}= \sqrt{25}= 5$ and so $\left(\sqrt{x}\right)^3= 5^3= 125$.

Generally,
$$\sqrt{x^3}= \left(x^3\right)^{1/2}= x^{3(1/2)}= x^{3/2}= \left(x^{1/2}\right)^2= \left(\sqrt{x}\right)^3$$

Even more generally,
$$\sqrt<b>{(x^a)}= \left(x^a\right)^{1/b}= x^{a/b}= \left(x^{1/b}\right)^a= \left(\sqrt<b>{x}\right)^a</b></b>$$

 

[eddysd]

Thanks HallsofIvy, couldn't have hoped for a better answer!

 

[Mute]

An important addition to Hall's post:

For certain values of x and some a and b the rule

$$\sqrt<b>{(x^a)}= \left(x^a\right)^{1/b}= x^{a/b}= \left(x^{1/b}\right)^a= \left(\sqrt<b>{x}\right)^a<br /> </b></b>$$

breaks down, and isn't true. For example, if x is negative and a = 2, b = 1/2. The rule implies
$$\sqrt{x^2} = (\sqrt{x})^2,$$
but this isn't true: $\sqrt{x}$ is not defined for x < 0 (on the set of reals, at least), and $\sqrt{(x^2)} = |x|$. So, you have to check that the expression is well defined before you can set them equal.

(There are other cases where the rule can break down, such as if you are using complex numbers, but you probably don't have to worry about that for now)

 

[Eldar]

Either one- they are equal. If x= 25, say, then $x^3= 25^3= 15625$ so $\sqrt{x^3}= \sqrt{15625}= 125$ while $\sqrt{x}= \sqrt{25}= 5$ and so $\left(\sqrt{x}\right)^3= 5^3= 125$.

Generally,
$$\sqrt{x^3}= \left(x^3\right)^{1/2}= x^{3(1/2)}= x^{3/2}= \left(x^{1/2}\right)^2= \left(\sqrt{x}\right)^3$$

Even more generally,
$$\sqrt<b>{(x^a)}= \left(x^a\right)^{1/b}= x^{a/b}= \left(x^{1/b}\right)^a= \left(\sqrt<b>{x}\right)^a</b></b>$$

You mistyped the 2nd bracket from the right of the line:

$$\sqrt{x^3}= \left(x^3\right)^{1/2}= x^{3(1/2)}= x^{3/2}= \left(x^{1/2}\right)^2= \left(\sqrt{x}\right)^3$$

It should be:

$$\left(x^{1/2}\right)^3$$

No doubt just a simple typing mistake.