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Chapter 3 of RD Sharma's Class 9 mathematics textbook focuses on the concept of rationalization, a crucial algebraic technique used to simplify expressions containing surds or irrational numbers. Exercise 3.2 specifically deals with the rationalization of denominators, which is an essential skill for simplifying complex fractions and solving equations involving irrational numbers. This set of problems aims to reinforce students' understanding of rationalization techniques and their application in various mathematical contexts.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Solution:
(i) We know that rationalisation factor for. We will multiply numerator and denominator of the given expression, to get
On equating rational and irrational terms, we get
Hence, we get a = 2, b = 1
(ii) We know that rationalisation factor for. We will multiply numerator and denominator of the given expression, to get
On equating rational and irrational terms, we get
Hence, we get a = 3, b = 2
(iii) We know that rationalisation factor for. We will multiply numerator and denominator of the given expression, to get
On equating rational and irrational terms, we get
Hence, we get a =, b =
(iv) We know that rationalisation factor for. We will multiply numerator and denominator of the given expression, to get
On equating rational and irrational terms, we get
Hence, we get a = -1, b = 1
(v) We know that rationalisation factor for. We will multiply numerator and denominator of the given expression, to get
On equating rational and irrational terms, we get
Hence, we get a =, b =
(vi) We know that rationalisation factor for. We will multiply numerator and denominator of the given expression, to get
On equating rational and irrational terms, we get
Hence, we get a =, b =
Solution:
We know that. We have to find the value of
As
Therefore,
We know that rationalization factor for. We will multiply numerator and denominator of the given expression
to get,
Putting the value of , we get
Hence the value of the given expression is 52.
Solution:
We know that . We have to find the value of
Therefore,
We know that rationalization factor for .
We will multiply numerator and denominator of the given expression
to get,
Putting the value of
We get,
Hence the given expression is simplified to 34.
Solution:
We know that for rationalization factor we will multiply denominator and numerator of the given expression
to get,
Putting the values of
we get,
Hence, value of the given expression is 11.904.
(i)
(ii)
Solution:
(i) We know that rationalization factor for
We will multiply numerator and denominator of the given expression
to get
Putting the values of
We get
Hence, the given expression is simplified to 0.102.
(ii) We know that rationalization factor for
We will multiply numerator and denominator of the given expression
to get
Putting the values of
We get
Hence, the given expression is simplified to 14.071.
Solution:
We have,
It can be simplified as
On squaring both sides, we get
The given equation can be rewritten as
Therefore, we have
Hence, the value of given expression is 10.
The practice questions in this set cover a range of rationalisation scenarios, including simple surds, compound surds, and expressions with multiple terms in the denominator. Students are challenged to rationalize denominators containing square roots, cube roots, and higher-order roots. Some problems involve algebraic manipulation alongside rationalisation, requiring students to apply multiple mathematical skills simultaneously. The questions progress in difficulty, starting with straightforward examples and advancing to more complex expressions that demand a deeper understanding of algebraic principles and rationalisation techniques.
