Question

What is the value of $$2ab(a+b)$$?

Answer 1

Hint: The distributive property of multiplication over addition is the property which states that multiplying a sum by a number gives identical result as multiplying the number(outside of the bracket) with each term present inside the bracket and then adding the product itself.
For example, we want to multiply $$5$$ with the sum $$(2+4)$$ then we get same result whether we multiply $$5$$ directly with the $$sum=6$$ i.e. $$5\times (2+4)=5\times 6=30$$or multiply 5 with each term present inside the bracket and then adding the product i.e. $$5\times 2+5\times 4=30$$

Complete step-by-step solution:
Step 1: Multiplying the number with each term present inside the bracket, we get
$$2ab(a+b)$$$$=2ab\times a+2ab\times b$$
Step 2: multiplying and adding the product, we get
$$\begin{gathered} 2ab(a+b)=2ab\times a+2ab\times b \\ 2ab(a+b)=2a^{2}b+2a{b}^2 \end{gathered}$$
This property is known as the distributive property of multiplication over addition.
Step 3: Proof of the property
Take an example of a number $$7$$ and a sum $$(5+9)$$and assume that these terms follow the above considered property
So we have,
$$7\times (5+9)=(7\times 5)+(7\times 9)$$
Taking, $$7\times (5+9)=L.H.S$$
$$L.H.S=7\times (5+9)$$
Adding the sum and then multiply it with the number, we get
$$\begin{gathered} L.H.S=7\times 14 \\ L.H.S=98 \end{gathered}$$
Again taking, $$(7\times 5)+(7\times 9)=R.H.S$$
$$R.H.S=(7\times 5)+(7\times 9)$$
Multiplying and adding the product, we get
$$\begin{gathered} R.H.S=35+63 \\ R.H.S=98 \end{gathered}$$
Hence, from above we have
$$L.H.S=R.H.S=98$$
Hence it is proved that,
$$2ab(a+b)=2a^{2}b+2a{b}^2$$

Note:
> Associative property explains that addition and multiplication of numbers are possible and does not depend on how they are grouped.
> Commutative property or commutative law explains that order of terms doesn’t matter while performing arithmetic operations. This property is applicable only for addition and multiplication processes.
> The distributive property tells us how to solve expressions in the form of a (b + c), it is also known as the distributive property of multiplication over addition.