TITLE: How to prove that $\sqrt{15}-\sqrt{8}+\sqrt[3]{7}<3$?
QUESTION [5 upvotes]: How to prove that $\sqrt{15}-\sqrt{8}+\sqrt[3]{7}<3$ without using calculus/high school methods or routine calculation of the roots with the precision needed? The difference between them is close enough to $0$, so trivial estimations like $\sqrt{15}<4$ or $\sqrt[3]{7}<2$ don't help here. What else can we try?

REPLY [3 votes]: Note that $15<(4-\frac 18)^2$ and $7<(2-\frac{1}{12})^3$, while $8>(3-\frac 15)^2$. Also, $\frac15-\frac18-\frac{1}{12}$ is negative. The result follows.