Question

How do you rewrite `1/(4b^3)` using negative exponents?

Answer 1

`4^(-1)b^(-3)`

Explanation:

Suppose we had `1/x`. This would have a particular value. If you turned it upside down to give `x/1` you have changed in inherent value so you need to apply a conversion.

The conversion used in this example is `x^(-1)`.

Although this looks deferent it has the same value as `1/x`

`color(green)("Note that "x" is the same as "x^1)`
so `1/x^1 = x^(-1)`

What if we had `1/(x^3)` then we would write: `x^(-3)`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Dealing with your question using the same logic.

Given : `1/(4b^3)`

`color(blue)("Assumption")`

`color(brown)(underline("Everything")" is meant to have negative exponents")`

Initially write it as: `1/4 xx1/b^3`

Now write as: `1/4 xx b^(-3)/1 = (1 xx b^(-3))/(4 xx 1)`

So now we have: `b^(-3)/4`

You could take this further by starting to 'play' with `1/4`

One potential format would be: `4^(-1)b^(-3)`

Another: another way of writing 4 is `2^2`

So we could write: `color(white)(..)4^(-1)b^(-3)color(white)(.)" as "color(white)(.)2^(-2)b^(-3)`

Which is equally true!