Simplify: $(a^{2} + 5) (b^{3} + 3) + 5$

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Solution

Step 1: Expand $(a^{2} + 5) (b^{3} + 3)$ using distributive property
Given: $(a^{2} + 5) (b^{3} + 3) + 5$
$∵ x (y + z) = x y + x z$
$\begin{array}{rcl} ∴ (a^{2} + 5) (b^{3} + 3) & = & a^{2} \times (b^{3} + 3) + 5 \times (b^{3} + 3) \\ = & (a^{2} \times b^{3}) + (a^{2} \times 3) + (5 \times b^{3}) + (5 \times 3) \\ = & a^{2} b^{3} + 3 a^{2} + 5 b^{3} + 15 \end{array}$
Step2: Find the expanded form of given expression
Now consider
$(a^{2} + 5) (b^{3} + 3) + 5$
$\begin{array}{rcl} = & a^{2} b^{3} + 3 a^{2} + 5 b^{3} + 15 + 5 \end{array}$
$\begin{array}{rcl} = & a^{2} b^{3} + 3 a^{2} + 5 b^{3} + 20 \end{array}$
Hence, $(a^{2} + 5) (b^{3} + 3) + 5 = a^{2} b^{3} + 5 b^{3} + 2 a^{2} + 20$

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