# Rules of powers

So I am working on an integral part of it involves $$\int (3t^2-4)^\frac{5}{2}$$

Obviously the sub is $$\Biggl[3\Biggl(\frac{2}{\sqrt{3}}sec(t)\Biggl)^2-4\Biggl]^{\frac{5}{2}}$$

which becomes

$$2\sqrt{tan^2(t)}^5$$ But how is that possible the $$5$$ is supposed to stay with whats inside the square root I fail to see where you can just pull out the 5 from inside the radical and say that its actually the radical and everything inside to the 5th power and not just what is inside the radical to the fifth power.

• It should be $$32\left(\sqrt{\tan^2 x}\right)^5$$ $32,$ not $2.$ Aug 27 '19 at 22:18

The property of exponents that matters is $$\left(a^b\right)^c=a^{bc}=\left(a^c\right)^b$$ for $$a \gt 0$$. If you have a square root sign, $$b$$ or $$c$$ is $$\frac 12$$ and this still applies. In fact, when you write $$(3t^2-4)^{\frac 52}$$ you are not indicating whether this is $$\sqrt{(3t^2-4)^5}$$ or $$\left(\sqrt{3t^2-4}\right)^5$$. It doesn't matter because they are equal by the preceding property.