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
\[2ab(a + b) = 2ab \times a + 2ab \times b \\
  2ab(a + b) = 2{a^2}b + 2a{b^2} \]
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
\[ L.H.S = 7 \times 14 \\
  L.H.S = 98 \]
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
\[ R.H.S = 35 + 63 \\
  R.H.S = 98 \]
Hence, from above we have
\[L.H.S = R.H.S = 98\]
Hence it is proved that,
\[2ab(a + b) = 2{a^2}b + 2a{b^2}\]

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.