I don't understand too well how this works
$\displaystyle \sqrt[3]{x+1=7}$
If I did that wrong please correct me . I saw it on my test today and did not understand it.
The question now is why should not have in mathematics formulas like:
$\displaystyle \sqrt[3]{x+1=7}$ and then taking to the cube the whole equation get an x=6??
Or even better solve 1st for x under the sqrt and then take 1/3 of 6 ??
Because functions, such as the square root, are defined on numbers, not equations. We sometimes talk about "taking the cube root" (or whatever other function you like) but what we mean is "take the cube root of each side".
You can for example, start with the equation, x+ 1= 7 and then take the cube root of both sides but that gives $\displaystyle \sqrt[3]{x+1}= \sqrt[3]{7}$, not $\displaystyle \sqrt[3]{x+ 1= 7}$.