Question

How do you simplify $\left( 5+\sqrt{3} \right)\left( 5-\sqrt{3} \right)$?

Answer 1

Hint: Now to find the value of given expression we will first multiply open the bracket by using distributive property. Now we will rearrange the terms and again use distributive property to expand the expression. Now we will simplify the expression by addition and subtraction of like terms. Hence we have the value of the given expression.

Complete step by step solution:
Now we want to find the value of given expression
To do so first let us first understand three basic properties of real numbers.
The first is commutativity.
Commutativity in addition means for any real number a and b $a+b=b+a$
Similarly Commutativity in multiplication means $a.b=b.a$ .
Now let us understand the property of associativity.
For any real numbers a, b and c associativity in addition means $a+\left( b+c \right)=\left( a+b \right)+c$ . Similarly associativity in multiplication means $a.\left( b.c \right)=\left( a.b \right).c$
Now finally let us understand the concept of distributive property.
Now for any real numbers a, b and c we have $a.\left( b-c \right)=ab-ac$ and $a.\left( b+c \right)=ab+ac$
Now consider the given expression $\left( 5+\sqrt{3} \right)\left( 5-\sqrt{3} \right)$
Now we consider $5+\sqrt{3}$ as one term.
Hence using distributive property we get,
$\Rightarrow \left( 5+\sqrt{3} \right)\left( 5 \right)-\left( 5+\sqrt{3} \right)\left( \sqrt{3} \right)$
Now using commutativity we get,
$\Rightarrow \left( 5 \right)\left( 5+\sqrt{3} \right)-\left( \sqrt{3} \right)\left( 5+\sqrt{3} \right)$
Now again using distributive property we get,
$\Rightarrow 5\left( 5 \right)+5\left( \sqrt{3} \right)-\left( \sqrt{3} \right)\left( 5 \right)-\left( \sqrt{3} \right)\left( \sqrt{3} \right)$
Now on simplifying the above equation we get,
$\begin{align}
  & \Rightarrow 25+5\sqrt{3}-5\sqrt{3}-3 \\
 & \Rightarrow 22 \\
\end{align}$
Hence we get the value of the given expression is 22.

Hence we get $\left( 5+\sqrt{3} \right)\left( 5-\sqrt{3} \right)=22$

Note: Now note that we can also find the value of the given expression without using the properties and expanding. For any real numbers a and b we have $\left( a-b \right)\left( a+b \right)={{a}^{2}}-{{b}^{2}}$ now the given expression is also in the form of $\left( a-b \right)\left( a+b \right)$ where $a=5$ and $b=\sqrt{3}$ . hence we can use the formula to find the value of the given expression.