Can cube root of x be expressed like this

If cube root of x can be expressed like x^1/3 then can't it also be expressed as x^-3 ?

Also, can x^-3 be expressed as -x^3 ?

∛x = x^(1/3)
x^(-3) = 1/x^3

No, two expression your wrote are mutually exclusive. x^1/3 is asking for the cube root of x while x^3 is simply multiplying x by itself 3 times.

2nd Q: once again no. X^3 is not equal to -x&3 because the negative sign in front of the second expression always yields a negative number.HOWEVER, if the value of x itself is negative, then yes, both expressions will be equaled.

But in maths, we can never assume what the value of x will be; thus, my conditional statement about probability of x being a negative number.

Yes, the cube root of x can be x^1/3.

Where you are confused, is thinking that x^1/3 = 1^-3. It does not.

x^-3 = 1/x^3. [Not the same thing as a cube root, I hope you can see that :) ]

Here's an example:

cube root 8 = 8^(1/3) = 2

8^-3 = 1/8^3 = 1/512 = 0.001953125

Not at all the same thing

1. No
x^-3 becomes 1/x^3

Also It cant be expressed as -x^3

No to both questions.

The roots are fractional powers.

The reciprocals are negative powers.