√2/(2 + √2). Rationalise the denominator and hence evaluate by taking √2 = 1.414, √3 = 1.732 and √5 = 2.236, upto three places of decimal
Solution:
Given, the expression is √2/(2+√2)
We have to rationalise the denominator and evaluate the given expression.
To rationalise we have to take conjugate,
√2/(2+√2) = √2/(2+√2) × (2-√2)/(2-√2)
= √2(2-√2) / (2-√2)(2+√2)
By using algebraic identity,
(a² - b²) = (a - b)(a + b)
(2-√2)(2+√2) = (2)² - (√2)²
= 4 - 2
= 2
So, √2(2-√2) / (2-√2)(2+√2) = √2(2-√2) / (2)
By multiplicative and distributive property,
√2(2-√2) = 2(√2) - √2(√2)
= 2√2 - 2
Now, √2(2-√2) / (2-√2)(2+√2) = (2√2 - 2) / (2)
Taking out common term,
= 2(√2 - 1)/2
= √2 - 1
So, √2/(2+√2) = √2 - 1
Given, √2 = 1.414
= 1.414 - 1
= 0.414
Therefore, √2/(2+√2) = 0.414
✦ Try This: Rationalise the denominator of the following and hence evaluate by taking √2 = 1.414, √3 = 1.732 and √5 = 2.236, upto three places of decimal √3/(3+√2)
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 1
NCERT Exemplar Class 9 Maths Exercise 1.3 Problem 13(iv)
√2/(2 + √2). Rationalise the denominator and hence evaluate by taking √2 = 1.414, √3 = 1.732 and √5 = 2.236, upto three places of decimal
Summary:
On rationalising the denominator and evaluating the expression √2/(2+√2) we get 0.414
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