Question

Simplify the given expression: $(5 + \sqrt 7 )(2 + \sqrt 5 )$

Answer 1

Hint: According to given in the question we have to simplify the given expression$(5 + \sqrt 7 )(2 + \sqrt 5 )$so first of all we have to multiply the first number/digit which is 5 with $(2 + \sqrt 5 )$and same as the second term/number which is $\sqrt 7 $with $(2 + \sqrt 5 )$and then we will add the whole expression obtained after multiplication and to multiply the square root terms we can use the formula as given below:
$
   \Rightarrow \sqrt a \times \sqrt b = \sqrt {ab} ................(1) \\
   \Rightarrow a \times \sqrt b = a\sqrt b ..............(2)
 $
Now, with the help of the formulas above we can obtain the values multiplication of square root terms.

Complete step-by-step answer:
Step 1: First of all we have to multiply the first number/digit which is 5 with $(2 + \sqrt 5 )$and same as the second term/number which is $\sqrt 7 $ with $(2 + \sqrt 5 )$
Hence,
$ = 5 \times (2 + \sqrt 5 ) + \sqrt 7 \times (2 + \sqrt 5 )$$…………..(3)$
Step 2: Now, we have to multiply each term of the expression obtained after multiplication with the help of the formulas (1) and (2) as mentioned in the solution hint.
$
   = 10 + 5\sqrt 5 + 2\sqrt 7 + \sqrt 7 \times \sqrt 5 \\
   = 10 + 5\sqrt 5 + 2\sqrt 7 + \sqrt {35}
 $

Hence, with the help of the formulas (1) and (2) we have simplified the given expression $(5 + \sqrt 7 )(2 + \sqrt 5 )$$ = 10 + 5\sqrt 5 + 2\sqrt 7 + \sqrt {35} $

Note: If we are multiplying two square root terms such as one term is $\sqrt a $ with another term $\sqrt a $ means both of the square root terms are the same then the product or value of their multiplication is a.
We can’t add the two different square root terms like if the given one term is $\sqrt a $ and the another term is $\sqrt b $