A) \[{{15}^{\frac{1}{4}}},\frac{{{3}^{\frac{1}{4}}}}{1}\times {{15}^{\frac{1}{8}}}\]
B) \[{{6}^{\frac{1}{2}}},{{2}^{\frac{1}{8}}}\times {{6}^{\frac{1}{4}}}\]
C) \[{{6}^{\frac{1}{8}}},{{2}^{\frac{1}{6}}}\times {{6}^{\frac{1}{6}}}\]
D) \[{{3}^{\frac{1}{8}}},{{2}^{\frac{1}{8}}}\times {{6}^{\frac{1}{8}}}\]
Correct Answer: A
Solution :
(a): We know that, it a and b are two distinct positive irrational numbers, then \[\sqrt{ab}\] is an irrational number lying between a and b. \[\therefore \]Irrational number between \[\sqrt{3}\] and \[\sqrt{5}\] is \[\sqrt{\sqrt{3}\times \sqrt{5}}=\sqrt{\sqrt{15}}={{15}^{1/4}}\] Irrational number between \[\sqrt{3}\] and \[{{15}^{1/4}}\] is\[\sqrt{\sqrt{3}\times {{15}^{1/4}}}={{3}^{1/4}}\times {{15}^{1/8}}\] Hence, required irrational numbers are \[{{15}^{\frac{1}{4}}}\] and \[{{3}^{\frac{1}{4}}}\times {{15}^{\frac{1}{8}}}\]
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