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.

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