Question

Simplify $(5 + \sqrt 5 )(5 - \sqrt 5 )$

Answer 1

Hint: We are given a mathematical expression $(5 + \sqrt 5 )(5 - \sqrt 5 )$ , the expression indicates the product of the sum of 5 and the square root of 5 and the difference of 5 and the square root of 5 (Square of a number means that number multiplied with itself, for example, 2 multiplied with itself gives 4, so 4 is the square of 2 and thus 2 is said to the square root of the number). The given expression can be solved by using an identity. We will use the identity that states, the product of the sum of two numbers and their difference is equal to the square of the first number minus the square of the second number, that is, $(a + b)(a - b) = {a^2} - {b^2}$ . Using the above-mentioned information, we can solve the given mathematical expression.

Complete step-by-step solution:
We have to simplify $(5 + \sqrt 5 )(5 - \sqrt 5 )$
We know that –
$
  (a + b)(a - b) = {a^2} - {b^2} \\
   \Rightarrow (5 + \sqrt 5 )(5 - \sqrt 5 ) = {(5)^2} - {(\sqrt 5 )^2} \\
   \Rightarrow (5 + \sqrt 5 )(5 - \sqrt 5 ) = 25 - 5 \\
   \Rightarrow (5 + \sqrt 5 )(5 - \sqrt 5 ) = 20 \\
 $
Hence, the simplified form of $(5 + \sqrt 5 )(5 - \sqrt 5 )$ is 20.

Note: The simplified form of an expression means a more understandable and easier way of writing the expression. We can find the value of $(5 + \sqrt 5 )(5 - \sqrt 5 )$ by multiplying each term in the first bracket with the whole second bracket one by one, that is, we will first multiply 5 with $(5 - \sqrt 5 )$ and then add it with the product of $\sqrt 5 $ and $(5 - \sqrt 5 )$ .
\[ \Rightarrow (5 + \sqrt 5 )(5 - \sqrt 5 ) = 5(5 - \sqrt 5 ) + \sqrt 5 (5 - \sqrt 5 )\]
The obtained expression can be further simplified by using the distributive property. According to the distributive property the product of one number with the sum of two other numbers is equal to the sum of the product of the number with the first number and the product of the number with second number, that is, $a(b + c) = ab + ac$
\[
   \Rightarrow 5(5 - \sqrt 5 ) + \sqrt 5 (5 - \sqrt 5 ) = 25 - 5\sqrt 5 + 5\sqrt 5 - 5 \\
   \Rightarrow (5 + \sqrt 5 )(5 - \sqrt 5 ) = 20 \\
 \]