Real Numbers
Class 9th Mathematics Part I MHB Solution
Practice Set 2.4
Question 1.Multiply
i. √3(√7 – √3)
ii. (√5 – √7)√2
iii. (3√2 – √3)(4√3 – √2)
Answer:i. √3(√7 – √3)
=√3 × √7 – √3 × √3
[∵√a(√b–√c)=√a×√b–√a×√c]
=√21 – 3
ii. (√5 – √7)√2
=√5 × √2 – √7 × √2
[∵√a(√b–√c)=√a×√b–√a×√c]
=√10 – √14
iii. (3√2 – √3)(4√3 – √2)
=3√2(4√3 – √2) – √3(4√3 – √2)
=3√2×4√3 – 3√2×√2 – √3×4√3 + √3×√2
[∵√a(√b–√c)=√a×√b–√a×√c]
=12√6 – 3×2 – 4×3 + √6
=12√6 – 6 – 12 + √6
=13√6 – 18
Question 2.Rationalize the denominator.
i.
ii. ![](data:image/png;base64,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)
iii.
iv. ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAXCAMAAADjjeWOAAAAAXNSR0ICQMB9xQAAAANQTFRFAAAAp3o92gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAADUlEQVQoz2NgGAX0BAABcAABXnnoOAAAAABJRU5ErkJggg==)
Answer:i. The rationalizing factor of √7 + √2 is √7 – √2. Therefore, multiply both numerator and denominator by √7 – √2.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
[∵ (a-b)(a+b) = a2 – b2]
![](data:image/png;base64,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)
![](data:image/png;base64,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)
ii. The rationalizing factor of 2√5 – 3√2 is 2√5 + 3√2. Therefore, multiply both numerator and denominator by 2√5 + 3√2.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
[∵ (a-b)(a+b) = a2 – b2]
![](data:image/png;base64,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)
![](data:image/png;base64,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)
iii. The rationalizing factor of 7 + 4√3 is 7 – 4√3. Therefore, multiply both numerator and denominator by 7 – 4√3.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOgAAAAxCAMAAADnRhWtAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOjoAOjqQOmaQOma2OpDbZgAAZgA6ZgBmZjoAZjpmZjqQZmZmZmaQZrb/kDoAkDo6kDpmkGY6kLbbkNv/tmYAtmY6ttv/tv//25A627Zm27aQ29u229v/2////7Zm/9uQ/9u2/9vb//+2///b+8t8EAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADLklEQVRoQ+2Z24KbIBCGYU/uyW6PtunartvdqJX3f78iKgIGxjBM03TjTXLB/M4ww4B8jJ2e0wwczwxU/HJ7PN7Geyq+PsYbH5Nl9/FN5JOx+uqY0oLwtfyy3rg6IynzONmKqyfQYCxd8e2JMfF8zfkFGESTkQQaKasy1GT+PNm6zY1couXZd9YWUBRd/gkasr445pEo2epc5mn34+hW93JY2a/TigNFXN7XFIFiZLu8d3/3o3V/v/YDfuiCLf2To5Tqmy1FoCjZwOqedNsN79PYPYypbze3P4OF1717ZASBomRF4d0ytO7Lq8rguLl0Oee3wf1UFLJIZKDVHk16xYrFydbe1WbqNpl0ffb7VxYs3Xpo5hxayCuCM4fgZEu9t4w6uoeYuqI4fzLPfzX3L+zRN4LSVas/ssepVAWeSVcmXm0u4wOZITwC0hwbKNg+xwkUxeWzmhJV0Mxf8NrPuCMMWM2RsqG9Rb1T61b8WrUW0Z8V2gL8WpOFH1lk4QqLlIUKYXa3y8f+024yzu/+46+Y6q18uYCr5zTgNANkM9B9GA6r7fhL9qJDC4vioo+0y+8O7Qn1++W2v5Vx/kP9s5yOxYl3XRnpy/urNJugdjHujy+pcWraapIVhXFDhNKkrj6kvhUoUiuFOWnpgufUFBEcVmNqRtGLFMlexutVtWQSzsRCVsXZby+xbRfJXpDmvqlZyrafh1Q24+/ek4pkL0hzn7sEskj2gjT3BUoga7MXGGjM3+3Kyd4cJhoUsrBmiL0sgYYo7AtOm2godAMTDQrZBOzFAhpOoA7RGNDNKqLBKGRR7MW+dHICdUiJQjdDDQNEQ24M5uVkGlkce7Ev5WyPZlJioxuQaBh3cmpe0sii2IsDNCyPNNGw0Q1MNPrIrJ0+iSyOvRj3u+6pdSYaNrqRGQKIhnVtnEwWx15moLGoMZNoOOhG3n+H+QCFLIq9uGTC3V4mUuKiG4BoUMhCECXMXtzu6Qu0RxgDullFNChkoUY/BbqTvSwa9iLQqdMZ5jDRoJDFsZcF0HADnYnGhG7YCqJBIfvX2ItGN3t/MQUNKGSR7AVp7guXSNZ83R+vBGfdyRIXZAAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
[∵ (a-b)(a+b) = a2 – b2]
![](data:image/png;base64,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)
![](data:image/png;base64,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)
iv. The rationalizing factor of √5 + √3 is √5 - √3. Therefore, multiply both numerator and denominator by √5 - √3.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
[∵ (a-b)(a+b) = a2 – b2]
![](data:image/png;base64,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)
[∵ (a-b)2 = a2 + b2 – 2ab]
![](data:image/png;base64,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)
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i. √3(√7 – √3)
ii. (√5 – √7)√2
iii. (3√2 – √3)(4√3 – √2)
i.
iii.