Question

How do you multiply ${\left( {3y + 2} \right)^2}?$

Answer 1

Hint: Use the law of indices for multiplication to split the given term into multiplicands or factors then consider one of the multiplicands in the multiplication to be constant and then use the distributive property of the multiplication to multiply the terms accordingly. You have to use the distributive property twice in this multiplication. Law of indices for multiplication is given as follows: ${a^m} = a \times a \times a \times a \times a.... \times m\;{\text{times}}$
And distributive property of multiplication is given as following: $a(b + c) = ab + ac$

Complete step by step solution:
 In order to multiply ${\left( {3y + 2} \right)^2}$ we will split the given term into factors with help of law of indices for multiplication to get the multiplicands at least which is must in multiplication. So using the law of indices for multiplication, we will get:
${\left( {3y + 2} \right)^2} = (3y + 2)(3y + 2)$
So we get our two factors or better say multiplicands to perform the multiplication,
Now multiplying them,
$(3y + 2) \times (3y + 2)$
With the help of distributive property of multiplication we can write it as
\[3y \times (3y + 2) + 2 \times (3y + 2)\]
Again applying the distributive property of multiplication, we will get
\[3y \times 3y + 3y \times 2 + 2 \times 3y + 2 \times 2\]
Simplifying it we will get
\[{(3y)^2} + 6y + 6y + 4 \\
\Rightarrow 9{y^2} + 6y + 6y + 4 \\ \]
With help of commutative property of addition, grouping similar terms, we will get
\[9{y^2} + \left( {6y + 6y} \right) + 4 \\
\therefore 9{y^2} + 12 + 4 \\ \]
Therefore the multiplication result of ${\left( {3y + 2} \right)^2}$ is equal to \[9{y^2} + 9y + 4\].

Note: Keep in mind that commutative property holds good only for addition and multiplication not for division and subtraction whereas distributive property is the same but in case of division it holds good when the numerator is distributed over the denominator. You can solve this by one more method, which is a direct algebraic formula and is given as following: ${(a + b)^2} = {a^2} + 2ab + {b^2}$