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# Why cant the square root of 4 be -2?

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Manager
Joined: 10 Dec 2011
Posts: 102
Location: India
Concentration: Finance, Economics
GMAT Date: 09-28-2012
WE: Accounting (Manufacturing)
Why cant the square root of 4 be -2? [#permalink]

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08 Jan 2018, 22:10
Why cannot the square root of 4 be -2?

When x^2 = 4, then x can be + or -2. But when x = (4)^1/2, then x = 2 only... Why?
-2 * -2 = 4.
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Math Expert
Joined: 02 Sep 2009
Posts: 46264
Re: Why cant the square root of 4 be -2? [#permalink]

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08 Jan 2018, 23:10
talismaaniac wrote:
Why cannot the square root of 4 be -2?

When x^2 = 4, then x can be + or -2. But when x = (4)^1/2, then x = 2 only... Why?
-2 * -2 = 4.

$$\sqrt{...}$$ is the square root sign, a function (called the principal square root function), which cannot give negative result. So, this sign ($$\sqrt{...}$$) always means non-negative square root.

The graph of the function f(x) = √x

Notice that it's defined for non-negative numbers and is producing non-negative results.

TO SUMMARIZE:
When the GMAT provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the non-negative root. That is:

$$\sqrt{9} = 3$$, NOT +3 or -3;
$$\sqrt[4]{16} = 2$$, NOT +2 or -2;

Notice that in contrast, the equation $$x^2 = 9$$ has TWO solutions, +3 and -3. Because $$x^2 = 9$$ means that $$x =-\sqrt{9}=-3$$ or $$x=\sqrt{9}=3$$.

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600px-Square_root_0_25.svg.png [ 11.45 KiB | Viewed 1755 times ]

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Manager
Joined: 10 Dec 2011
Posts: 102
Location: India
Concentration: Finance, Economics
GMAT Date: 09-28-2012
WE: Accounting (Manufacturing)
Re: Why cant the square root of 4 be -2? [#permalink]

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08 Jan 2018, 23:34
Bunuel wrote:

TO SUMMARIZE:
When the GMAT provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the non-negative root. That is:

$$\sqrt{9} = 3$$, NOT +3 or -3;
$$\sqrt[4]{16} = 2$$, NOT +2 or -2;

Notice that in contrast, the equation $$x^2 = 9$$ has TWO solutions, +3 and -3. Because $$x^2 = 9$$ means that $$x =-\sqrt{9}=-3$$ or $$x=\sqrt{9}=3$$.

I think I do understand some of this. Much Thanks!!
Manhattan Prep Instructor
Joined: 04 Dec 2015
Posts: 529
GMAT 1: 790 Q51 V49
GRE 1: 340 Q170 V170
Re: Why cant the square root of 4 be -2? [#permalink]

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14 Jan 2018, 14:25
Bunuel wrote:
talismaaniac wrote:
Why cannot the square root of 4 be -2?

When x^2 = 4, then x can be + or -2. But when x = (4)^1/2, then x = 2 only... Why?
-2 * -2 = 4.

Check this out: https://www.manhattanprep.com/gmat/blog ... -the-gmat/

The short explanation is that it's somewhat arbitrary - there are mathematical reasons for thinking about it that way that aren't really important on the GMAT. It's one of those things that you can (and should) just memorize, since the GMAT will be very predictable. But the article also explains it a bit.
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Re: Why cant the square root of 4 be -2?   [#permalink] 14 Jan 2018, 14:25
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