What is the value of 3(root5(root3(...), where the given expression

Question

What is the value of  3\sqrt{5\sqrt{3\sqrt{5\sqrt{...}}}} , where the given expression extends to an infinite number of roots?

A.  3\sqrt{5}

B.  3\sqrt {75}

C. 15

D.  3\sqrt{75}

E. 75

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Bunuel · Answer

GMATinsight · Answer

Let,  3\sqrt{5\sqrt{3\sqrt{5\sqrt{...}}}} = x 
i.e.  3\sqrt{5\sqrt{x}} = x

Squaring both sides

i.e.  9*5\sqrt{x} = x^2

Squaring both sides AGAIN

i.e.  9^2*5^2*x = x^4

i.e.  3^3*3*5^2 = x^3

Taking cube root both sides

i.e.  3\sqrt {75} = x^3

Answer: Option B

yashikaaggarwal · Answer

Let 3√5√3√5√3............∞ = Y
3√5√Y = Y
Squaring both sides
9*5√Y = Y^2
Squaring again
81*25*Y = Y^4
3*3*3*3*5*5 = Y^3
Y = 3³√75

Answer is B

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Kinshook · Answer

What is the value of  3\sqrt{5\sqrt{3\sqrt{5\sqrt{...}}}} , where the given expression extends to an infinite number of roots?

x^2 = 9×5\sqrt{x} 
 x^4 = 81×25x 
 x^3 = 3^3×75 
 x = 3\sqrt {75}

IMO B

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BrushMyQuant · Answer

↧↧↧ Detailed Video Solution to the Problem Series ↧↧↧

https://www.youtube.com/watch?v=6Cbkvz4PsLM

3\sqrt{5\sqrt{3\sqrt{5\sqrt{...}}}}

Let,  3\sqrt{5\sqrt{3\sqrt{5\sqrt{...}}}}  = x

Note that the terms is a sequence of 3 and 5 under square roots and is extending till infinity, so we can replace the part of the term after first 3 and 5 by x itself

=>  3\sqrt{5\sqrt{3\sqrt{5\sqrt{...}}}}  =  3\sqrt{5\sqrt{x}}

Squaring both the sides we get 
=>  3^2 * (\sqrt{5\sqrt{x}})^2 = x^2 
=>  3^2 * 5\sqrt{x} = x^2

Squaring both the sides again, we get 
 =>  3^4 * 5^2 * (\sqrt{x})^2 = x^4 
 =>  3^4 * 5^2 * x = x^4 
 =>  x^3 = 3^4 * 5^2 
 =>  x^3 = 3^3 * 3 * 25 = 3^3 * 75

Taking cube root on both the sides we get 
=> x =  3\sqrt {75}

So,  Answer will be B 
Hope it helps!

Watch the following video to MASTER Roots

https://www.youtube.com/watch?v=5oAec-Y6kOQ

HarshavardhanR · Answer

https://gmatclub.com/forum/download/file.php?id=83406 Harsha GMAT-Club-Forum-hmuhsnce.png

What is the value of 3(root5(root3(...), where the given expression

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What is the value of 3(root5(root3(...), where the given expression

Prep Toolkit

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