Differentials to find approximate values

Question

I'm asked to solve the following without a calculator: $80^{3/4}$

I only know that $f(x+dx) \approx f(x) + dy$

I then proceed to find $dy$, it should follow that if $f(x) = x^{3/4}$, then $dy = \dfrac{3}{4\sqrt[4]{x}}dx$.

The issue I have at this point is the following: I know that $81^{3/4} = 27$ so it would be very useful to have $dx = -1$ but there is no $\sqrt[4]{-1}$ (even if there is,at this point I am not supposed to know complex numbers).

How do I proceed? What is that I am missing?

Answer 1

You are substituting $-1$ for $x$ when you should be substituting $-1$ for $dx$. In this case $x = 81$.

Answer 2

If you confused between $x$ and $dx$ you have not yet come to grips with central concept of differential calculus.

$$ dy = \dfrac{3}{4\sqrt[4]{x}}dx = \dfrac{3}{4\sqrt[4]{81}}(-1). $$