Question

How do you use multiplication to solve ${{\left( 4a+b \right)}^{2}}$

Answer 1

Hint: Now to solve the expression we will first write the square in multiplication form. Now let us open the brackets by using the distributive property. We know that according to distributive property we have $a\left( b+c \right)=a.b+a.c$ . Hence using this bracket we will simplify the given expression.

Complete step by step solution:
Now let us first understand three basic properties of real number
First let us understand commutativity.
Now for addition we have $a+b=b+a$ similarly for multiplication we have $ab=ba$ .
Now similarly we have associativity
For addition we have \[\left( a+b \right)+c=a+\left( b+c \right)\] and also for multiplication we have $\left( a.b \right).c=a.\left( b.c \right)$ .
Now let us understand the distributive property.
$a.\left( b+c \right)=a.b+a.c$
Now let us consider the given expression ${{\left( 4a+b \right)}^{2}}$ ?
Now we know that ${{a}^{2}}$ is nothing but $a\times a$
Hence first we will write the expression in multiplication format.
Hence we write the given expression as $\left( 4a+b \right)\times \left( 4a+b \right)$ .
Now we know that according to distributive property we have $a\left( b+c \right)=ab+ac$
Hence using this property we get,
$\Rightarrow \left( 4a+b \right)\left( 4a \right)+\left( 4a+b \right)\left( b \right)$
Now we will use the commutative property of multiplication which says $a\times b=b\times a$. Hence we get,
$\Rightarrow \left( 4a \right)\left( 4a+b \right)+\left( b \right)\left( 4a+b \right)$
Now again using distributive property and open the bracket hence we get,
$\Rightarrow 16{{a}^{2}}+4ab+4ab+{{b}^{2}}$
Now simplifying the equation we get,
$\begin{align}
  & \Rightarrow 16{{a}^{2}}+\left( 4+4 \right)ab+{{b}^{2}} \\
 & \Rightarrow 16{{a}^{2}}+8ab+{{b}^{2}} \\
\end{align}$
Hence we get, ${{\left( 4a+b \right)}^{2}}=16{{a}^{2}}+8ab+{{b}^{2}}$
Hence the multiplication of the given term is $16{{a}^{2}}+8ab+{{b}^{2}}$ .

Note:
Now note that the distributive property is given as $a\left( b+c \right)=ab+ac$ and also we can say $\left( a+b \right)c=ac+bc$ . Hence we can expand the term on both sides. Now also note that we have direct formula to expand the term ${{\left( a+b \right)}^{2}}$ and the square is given by ${{a}^{2}}+2ab+{{b}^{2}}$ .
Hence we have ${{\left( 4a+b \right)}^{2}}={{\left( 4a \right)}^{2}}+2\left( 4a \right)\left( b \right)+{{b}^{2}}=16{{a}^{2}}+8ab+{{b}^{2}}$ .