How do you simplify `(5x ^ { 3} ) ( 2x ) ^ { - 3}`?

When you see some term raised to a negative power, it's really shorthand for division.

In other words, `a^-n = 1 / (a^n)`.

So `(5x^3)(2x)^(-3) = (5x^3)/(2x)^3`.

Notice that the `2` in the denominator is inside the parentheses, so we must raise it to the third power.

`(5x^3) / (2x)^3 = (5x^3) / (8x^3)`.

At this point, we notice that `x^3` is in the numerator and denominator, so it can be canceled. (Removed.)

`(5x^3)/(8x^3) = 5 / 8`.

And `5/8` cannot be further simplified, so this is our final answer.

Note: It is also appropriate to mention that `x ne 0`, since if `x = 0` then `(5x^3) / (8x^3) = 0 / 0` and one cannot divide by zero.

First, let's rewrite this without the negative exponent. The negative means that the term is in the denominator:

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

Now, let's distribute the power in the denominator:

`(5x^3)/(2^3x^3)=(5x^3)/(8x^3)`

The last step is to cancel out the `x^3` terms which gives us:

`5/8`
We can do this because any number divided by itself is 1 which means:

`(5x^3)/(8x^3)=5/8*x^3/x^3= 5/8*1`

We can also approach the original problem a different way.
After we distribute the `-3` exponent, we get the following:

`(5x^3)(2x)^-3=(5x^3)(2^-3)(x^-3)`

Now when we simplify this equation we get:

`5*x^3*1/8*x^-3`

After rearranging this expression we can then use the rules of exponents. When you multiply terms with the same base you add their exponents together:

`5*1/8*x^3*x^-3=5/8x^(3+(-3))=5/8x^0=5/8`