If $a$ and $b$ are positive numbers, which of the following is equivalent to $\sqrt{(a+b)^3}\cdot\sqrt{a+b}$ ?

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We can rewrite the expression with fractional exponents.

$$\sqrt{(a+b)^3}\cdot\sqrt{a+b}$$ $$=(a+b)^{\frac{3}{2}}\cdot (a+b)^{\frac{1}{2}}$$ $$= (a+b)^{\frac{3}{2}+\frac{1}{2}}$$ $$= (a+b)^2$$ $$= \boxed{a^2+2ab+b^2}$$

We can combine the radicals into one.

$$\sqrt{(a+b)^3}\cdot\sqrt{a+b}$$ $$= \sqrt{(a+b)^3 \cdot (a+b)}$$ $$=\sqrt{(a+b)^4}$$ $$=\sqrt{(a+b)^2 \cdot (a+b)^2}$$ $$= (a+b)^2$$ $$= \boxed{a^2+2ab+b^2}$$