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# If x = 4 and y = 6, then (x^(1/2) + y^(1/2))/(x+y)^(1/2)

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Math Expert
Joined: 02 Sep 2009
Posts: 43864
If x = 4 and y = 6, then (x^(1/2) + y^(1/2))/(x+y)^(1/2) [#permalink]

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05 Sep 2017, 00:19
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35% (medium)

Question Stats:

66% (01:40) correct 34% (01:15) wrong based on 137 sessions

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If x = 4 and y = 6, then $$\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x+y}}=$$

A. $$\frac{\sqrt{10}+\sqrt{15}}{10}$$

B. 1

C. $$\frac{2\sqrt{5}}{10}$$

D. $$\frac{\sqrt{10}+\sqrt{15}}{5}$$

E. 2
[Reveal] Spoiler: OA

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If x = 4 and y = 6, then (x^(1/2) + y^(1/2))/(x+y)^(1/2) [#permalink]

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05 Sep 2017, 02:39
2
KUDOS
We have been asked to find the value of the expression $$\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x+y}}$$

Multiplying and dividing the expression by $$\sqrt{10}$$ does not change the value of the expression

The expression becomes $$\frac{\sqrt{10x}+\sqrt{10y}}{\sqrt{10(x+y)}}$$
= $$\frac{\sqrt{2*5*2*2}+\sqrt{2*5*2*3}}{\sqrt{10(4+6)}}$$ (Substituting x=4 and y=6)

= $$\frac{2\sqrt{5*2}+2\sqrt{5*3}}{10}$$ = $$\frac{\sqrt{10}+\sqrt{15}}{5}$$ (Option D)
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Re: If x = 4 and y = 6, then (x^(1/2) + y^(1/2))/(x+y)^(1/2) [#permalink]

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30 Sep 2017, 07:47
pushpitkc wrote:
We have been asked to find the value of the expression $$\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x+y}}$$

Multiplying and dividing the expression by $$\sqrt{10}$$ does not change the value of the expression

The expression becomes $$\frac{\sqrt{10x}+\sqrt{10y}}{\sqrt{10(x+y)}}$$
= $$\frac{\sqrt{2*5*2*2}+\sqrt{2*5*2*3}}{\sqrt{10(4+6)}}$$ (Substituting x=4 and y=6)

= $$\frac{2\sqrt{5*2}+2\sqrt{5*3}}{10}$$ = $$\frac{\sqrt{10}+\sqrt{15}}{5}$$ (Option D)

hi

"Multiplying and dividing the expression by √10 does not change the value of the expression" that's okay ..but can you please tell me why the following method - squaring the whole expression - is not working here

(√x + √y) / √(x + y)

= { (√x + √y) / √(x + y) } ^ 2

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Joined: 22 May 2017
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Re: If x = 4 and y = 6, then (x^(1/2) + y^(1/2))/(x+y)^(1/2) [#permalink]

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30 Sep 2017, 08:10
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Plug in the values
√4 + √6 / √(4+6)
=2+√6 / √10

Now you see there is no option with this ans
Observe the options - fraction choices have denominator without root

Hence to remove the root multiply the numerator & denominator with √10

=(2 + √6) (√10) / (√10)(√10)
=2√10+√60 / 10
=2√10+√(4*15) /10
=2√10 + 2√15 / 10
Take common 2 out on numerator and dividing leaves denominator as 5
=√10+√15 / 5

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If x = 4 and y = 6, then (x^(1/2) + y^(1/2))/(x+y)^(1/2) [#permalink]

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30 Sep 2017, 08:11
2
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Quote:
hi

"Multiplying and dividing the expression by √10 does not change the value of the expression" that's okay ..but can you please tell me why the following method - squaring the whole expression - is not working here

(√x + √y) / √(x + y)

= { (√x + √y) / √(x + y) } ^ 2

Hi gmatcracker2017
it will work but the problem is the options are not in their simplest form. Hence you will have to undertake lot of calculation to arrive at the answer choice.

Straight forward Algebric method would be -
$$(\sqrt{x}+\sqrt{y})/\sqrt{x+y}$$, multiply the numerator and denominator by $$\sqrt{x+y}$$ to get

$$(\sqrt{x}+\sqrt{y})(\sqrt{x+y})/x+y$$ $$=(\sqrt{x(x+y)}+\sqrt{y(x+y)})/x+y$$. Now substitute the values of $$x$$ & $$y$$ to get

$$(\sqrt{40}+\sqrt{60})/10$$ $$=(\sqrt{10}+\sqrt{15})/5$$

Option D
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Re: If x = 4 and y = 6, then (x^(1/2) + y^(1/2))/(x+y)^(1/2) [#permalink]

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30 Sep 2017, 08:35
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niks18 wrote:
Quote:
hi

"Multiplying and dividing the expression by √10 does not change the value of the expression" that's okay ..but can you please tell me why the following method - squaring the whole expression - is not working here

(√x + √y) / √(x + y)

= { (√x + √y) / √(x + y) } ^ 2

Hi gmatcracker2017
it will work but the problem is the options are not in their simplest form. Hence you will have to undertake lot of calculation to arrive at the answer choice.

Straight forward Algebric method would be -
$$(\sqrt{x}+\sqrt{y})/\sqrt{x+y}$$, multiply the numerator and denominator by $$\sqrt{x+y}$$ to get

$$(\sqrt{x}+\sqrt{y})(\sqrt{x+y})/x+y$$ $$=(\sqrt{x(x+y)}+\sqrt{y(x+y)})/x+y$$. Now substitute the values of $$x$$ & $$y$$ to get

$$(\sqrt{40}+\sqrt{60})/10$$ $$=(\sqrt{10}+\sqrt{15})/5$$

Option D

hi niks18

thanks a lot, man
I got it

Re: If x = 4 and y = 6, then (x^(1/2) + y^(1/2))/(x+y)^(1/2)   [#permalink] 30 Sep 2017, 08:35
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