Question

The Arithmetic mean of two surds is
5 + 4√3
and one of them is
2 + 4√3
then the
square root of the second surd is​

Answer 1

Step-by-step explanation:

Given :-

The Arithmetic mean of two surds is

5 + 4√3 and one of them is 2 + 4√3

To find :-

Find the square root of the second ssurd?

Solution :-

Given that :

The Arithmetic mean of two surds = 5 + 4√3

One of the surds = 2 + 4√3

Let the other surd be X

Arithmetic Mean of the two numbers a and b is (a+b)/2

=> The Arithmetic mean of (2+4√3) and X

=> (2+4√3+X)/2

According to the given problem

The Arithmetic mean of two surds = 5 + 4√3

=> (2+4√3+X)/2 = 5 + 4√3

=> (2+4√3+X) = 2(5+4√3)

=> 2+4√3+X = 10+8√3

=> X = (10+8√3)-(2+4√3)

=>X = 10+8√3-2-4√3

=> X = (10-2) +(8√3-4√3)

=>X = 8+4√3

The other surd = 8+4√3

The square root of 8+4√3

=>√(8+4√3)

=> √(8+2(√3×2×2))

=>√(8+2√12)

=> √[(6+2)+2(√6×√2)]

=>√[(√6)^2+(√2)^2+2(√6×√2)]

It is in the form of a^2+2ab+b^2

Where a = √6 and b=√2

We know that

a^2+2ab+b^2 = (a+b)^2

=> √[(√6)^2+(√2)^2+2(√6×√2)]

=>√[(√6+√2)^2]

=>√6+√2

Therefore, √(8+4√3) = √6+√2

Answer:-

The square root of 8+4√3= √6+√2

Used formulae:-

• Arithmetic Mean of the two numbers a and b is (a+b)/2

• (a+b)^2 = a^2+2ab+b^2