Simplify : 7√3/(√10+√3) - 2√5/(√6+√5) - 3√2/(√15+3√2)
Solution:
Given, the expression is 7√3/(√10+√3) - 2√5/(√6+√5) - 3√2/(√15+3√2)
We have to simplify the expression.
Considering 7√3/(√10+√3),
By taking conjugate,
7√3/(√10+√3) = 7√3/(√10+√3) × (√10-√3)/(√10-√3)
= 7√3(√10-√3) / (√10+√3)(√10-√3)
By using algebraic identity,
(a - b)(a + b) = a² - b²
(√10+√3)(√10-√3) = (√10)² - (√3)²
= 10 - 3
= 7
So, 7√3(√10-√3) / (√10+√3)(√10-√3) = 7√3(√10-√3)/7
= √3(√10-√3)
7√3/(√10+√3) = √30 - 3
Considering 2√5/(√6+√5),
By taking conjugate,
2√5/(√6+√5) = 2√5/(√6+√5) × (√6-√5)/(√6-√5)
= 2√5(√6-√5) / (√6+√5)(√6-√5)
By using algebraic identity,
(a - b)(a + b) = a² - b²
(√6-√5)(√6+√5) = (√6)² - (√5)²
= 6 - 5
= 1
So, 2√5(√6-√5) / (√6+√5)(√6-√5) = 2√5(√6-√5)/(1)
= 2√5(√6-√5)
= 2√5(√6) - 2√5(√5)
= 2√30 - 2(5)
2√5/(√6+√5) = 2√30 - 10
Considering 3√2/(√15+3√2),
By taking conjugate,
3√2/(√15+3√2) = 3√2/(√15+3√2) × (√15-3√2)/(√15-3√2)
= 3√2(√15-3√2) / (√15+3√2)(√15-3√2)
By using algebraic identity,
(a - b)(a + b) = a² - b²
(√15-3√2)(√15+3√2) = (√15)² - (3√2)²
= 15 - 18
= -3
So, 3√2(√15-3√2) / (√15+3√2)(√15-3√2) = 3√2(√15-3√2) / (-3)
= (-1)√2(√15-3√2)
= -√2(√15) + 3√2(√2)
= 3(2) - √30
3√2/(√15+3√2) = 6 - √30
Now, 7√3/(√10+√3) - 2√5/(√6+√5) - 3√2/(√15+3√2) = √30 - 3 - (2√30 - 10) - (6 - √30)
= √30 - 3 - 2√30 + 10 - 6 + √30
= √30 - 2√30 + √30 - 3 - 6 + 10
= 2√30 - 2√30 - 9 + 10
= 1
Therefore, 7√3/(√10+√3) - 2√5/(√6+√5) - 3√2/(√15+3√2) = 1
✦ Try This: Simplify : 8√3/(√12+√2) - 3√5/(√8+√2)
NCERT Exemplar Class 9 Maths Exercise 1.4 Problem 2
Simplify : 7√3/(√10+√3) - 2√5/(√6+√5) - 3√2/(√15+3√2)
Summary:
The simplified value of the expression 7√3/(√10+√3) - 2√5/(√6+√5) - 3√2/(√15+3√2) is 1
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