Simplifying indices with surds

Question

I am asked to simplify $(\sqrt{t^3}) \times (\sqrt{t^5})$.

I get up to $\sqrt[3]{t^3}\times \sqrt{t^5}$ but I am not sure how to simplify this further as now roots are involved and not just powers.

When I checked the solutions the final answer should be $t^4$ but I'm not sure how this is achieved.

Answer 1

One way is to note that $\left( \sqrt t \right)^3=t^{\frac 32}$ and similarly for the other one. Then when you multiply terms you add exponents

Answer 2

If my edit is correct you have: $$ \sqrt{t^3}\times \sqrt{t^5}=\sqrt{t^3\times t^5 }=\sqrt{t^8}=t^4 $$

or, with fractional exponents: $$ \sqrt{t^3}\times \sqrt{t^5}=t^{\frac{3}{2}}t^{\frac{5}{2}}=t^{\frac{3}{2}+\frac{5}{2}}=t^{\frac{8}{2}}=t^4 $$