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03 Jan 2024, 09:03
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E
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Difficulty:
5%
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Question Stats:
88% (01:08) correct
12%
(01:42)
wrong
based on 304
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Date
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\(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} = \)
A. 2
B. 4
C. \(\sqrt{15}\)
D. \(8\sqrt{15}\)
E. \(4 + \sqrt{15}\)
2024-01-03_19-33-12.png [ 9.16 KiB | Viewed 3750 times ]
A. 2
B. 4
C. \(\sqrt{15}\)
D. \(8\sqrt{15}\)
E. \(4 + \sqrt{15}\)
Attachment:
2024-01-03_19-33-12.png [ 9.16 KiB | Viewed 3750 times ]
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03 Jan 2024, 09:10
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guddo
Rationalize the fraction by multiplying both the numerator and the denominator by \(\sqrt{5} + \sqrt{3}\). The purpose of rationalization is to eliminate irrational expressions in the denominator and simplify the expression:
- \(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} =\frac{(\sqrt{5} + \sqrt{3})(\sqrt{5} + \sqrt{3})}{(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})}\)
Apply \((a + b)^2 = a^2 + 2ab + b^2\) to the numerator and \((a-b)(a+b) = a^2-b^2\) to the denominator:
- \(\frac{5 + 2\sqrt{15} + 3}{5 -3} = \)
\(=\frac{8 + 2\sqrt{15}}{2} = \)
\(=4 + \sqrt{15}\).
Answer: E.
25 Jul 2025, 22:10
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We need to simplify \(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}\)
We will be using a concept called as Rationalizing Roots, we will multiply the numerator and the denominator with the conjugate of the denominator
Conjugate of \(\sqrt{5} - \sqrt{3}\) = \(\sqrt{5} + \sqrt{3} \)
=> \(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}\) = \(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}\) * \(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}\)
=> \(\frac{(\sqrt{5} + \sqrt{3})^2}{(\sqrt{5} - \sqrt{3})* (\sqrt{5} + \sqrt{3})}\)
=> \(\frac{(\sqrt{5})^2 + 2*\sqrt{5}*\sqrt{3} + (\sqrt{3})^2}{(\sqrt{5})^2 - (\sqrt{3})^2}\)
=> \(\frac{5 + 2*\sqrt{15} + 3}{5-3}\)
=> \(\frac{8 + 2\sqrt{15}}{2}\)
=> \(4 + \sqrt{15}\)
So, Answer will be E
Hope it helps!
Playlist on Solved Problems on Roots here
Watch the following video to MASTER Rationalizing Roots
We will be using a concept called as Rationalizing Roots, we will multiply the numerator and the denominator with the conjugate of the denominator
Conjugate of \(\sqrt{5} - \sqrt{3}\) = \(\sqrt{5} + \sqrt{3} \)
=> \(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}\) = \(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}\) * \(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}\)
=> \(\frac{(\sqrt{5} + \sqrt{3})^2}{(\sqrt{5} - \sqrt{3})* (\sqrt{5} + \sqrt{3})}\)
=> \(\frac{(\sqrt{5})^2 + 2*\sqrt{5}*\sqrt{3} + (\sqrt{3})^2}{(\sqrt{5})^2 - (\sqrt{3})^2}\)
=> \(\frac{5 + 2*\sqrt{15} + 3}{5-3}\)
=> \(\frac{8 + 2\sqrt{15}}{2}\)
=> \(4 + \sqrt{15}\)
So, Answer will be E
Hope it helps!
Playlist on Solved Problems on Roots here
Watch the following video to MASTER Rationalizing Roots
