Question

Simplify the below given expression
\[\left( 5+\sqrt{5} \right)\left( 5-\sqrt{5} \right)\]

Answer 1

Hint: To solve this question we will use the product rule of multiplication of two bracket terms. When the terms are given as (a – b) and (a + b), then there is an identity to solve them. It is given as, \[\left( a-b \right)\left( a+b \right)={{a}^{2}}-{{b}^{2}}\]
Using this identity, we will solve this question.

Complete step-by-step solution:
We are given our expression as
\[\left( 5+\sqrt{5} \right)\left( 5-\sqrt{5} \right)......\left( i \right)\]
There is an identity relating (a + b) and (a – b) given below.
\[\left( a-b \right)\left( a+b \right)={{a}^{2}}-{{b}^{2}}\]
Comparing the given equation (i) and comparing to the identity stated above, we have,
\[a=5\]
\[b=\sqrt{5}\]
Now, let us write the term (a - b) by substituting them as shown below,
\[\Rightarrow \left( a-b \right)=\left( 5-\sqrt{5} \right)\]
Now, let us write the term (a + b) by substituting them as shown below,
\[\Rightarrow \left( a+b \right)=\left( 5+\sqrt{5} \right)\]
Now, we can express them as a product as below,
\[\left( a-b \right)\left( a+b \right)=\left( 5+\sqrt{5} \right)\left( 5-\sqrt{5} \right)\]
As we know by the above stated identity that,
\[\left( a-b \right)\left( a+b \right)={{a}^{2}}-{{b}^{2}}\]
Applying this, we have,
\[\left( 5+\sqrt{5} \right)\left( 5-\sqrt{5} \right)={{\left( 5 \right)}^{2}}-{{\left( \sqrt{5} \right)}^{2}}\]
Simplifying the terms and writing their values, we have
\[\Rightarrow \left( 5+\sqrt{5} \right)\left( 5-\sqrt{5} \right)=25-5\]
Simplifying the terms, we have
\[\Rightarrow \left( 5+\sqrt{5} \right)\left( 5-\sqrt{5} \right)=20\]
Therefore, the value of \[\left( 5+\sqrt{5} \right)\left( 5-\sqrt{5} \right)\] is given as 20.

Note: Another method to solve this question can be by without applying any identity and directly solving by opening the bracket. This method is as below.
\[\left( 5+\sqrt{5} \right)\left( 5-\sqrt{5} \right)=5\left( 5-\sqrt{5} \right)+\sqrt{5}\left( 5-\sqrt{5} \right)\]
Again, we will multiply each term by opening the brackets and we will get
\[\Rightarrow \left( 5+\sqrt{5} \right)\left( 5-\sqrt{5} \right)={{5}^{2}}-5\sqrt{5}+5\sqrt{5}-\left( \sqrt{5} \right)\left( \sqrt{5} \right)\]
Now, cancelling similar terms, we will get
\[\Rightarrow \left( 5+\sqrt{5} \right)\left( 5-\sqrt{5} \right)=25-5\]
\[\Rightarrow \left( 5+\sqrt{5} \right)\left( 5-\sqrt{5} \right)=20\]
Therefore, the answer we obtained here is the same answer we got before.