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Bunuel
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If (x^2)^(1/2)/(x^3)^(1/3) = -1and x 0, what is the value of 26 Apr 2024, 01:30
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68% (01:27) correct 32% (01:28) wrong based on 294 sessions

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­If \(\frac{\sqrt{x^2}}{\sqrt[3]{x^3}}=-1\) and \(x \neq 0\), what is the value of \(\frac{2|x| + 6}{3 - x}\) ?

A. -2
B. 1
C. 2
D. 3
E. Cannot be determined from the given information­

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Bunuel
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If (x^2)^(1/2)/(x^3)^(1/3) = -1and x 0, what is the value of 05 May 2024, 04:45
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Official Solution:

If \(\frac{\sqrt{x^2}}{\sqrt[3]{x^3}}=-1\) and \(x \neq 0\), what is the value of \(\frac{2|x| + 6}{3 - x}\) ?

A. -2
B. 1
C. 2
D. 3
E. Cannot be determined from the given information­


\(\frac{\sqrt{x^2}}{\sqrt[3]{x^3}} = -1\) simplifies to \(\frac{|x|}{x} = -1\), which then yields \(|x| = -x\). This indicates that \(x\) must be negative. If \(x < 0\), then \(|x| = -x\), and thus we get \(\frac{-2x + 6}{3 - x} = \frac{2(3 - x)}{3 - x} = 2\).


Answer: C­
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If (x^2)^(1/2)/(x^3)^(1/3) = -1and x 0, what is the value of 26 Apr 2024, 02:18
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Bunuel
­If \(\frac{\sqrt{x^2}}{\sqrt[3]{x^3}}=-1\) and \(x \neq 0\), what is the value of \(\frac{2|x| + 6}{3 - x}\) ?

A. -2
B. 1
C. 2
D. 3
E. Cannot be determined from the given information­
­
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\(\frac{\sqrt{x^2}}{\sqrt[3]{x^3}}\)

= \(\frac{|x|}{x}\)

\(\frac{|x|}{x} = -1\)

Hence, we can infer that that \(x\) is negative. 
­
Let's assume \(x = -1\)

= \(\frac{2|x| + 6}{3 - x}\)

= \(\frac{2|-1| + 6}{3 - (-1)}\)

= \(\frac{8}{4}\)

= \(2\)

­Option C
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If (x^2)^(1/2)/(x^3)^(1/3) = -1and x 0, what is the value of 27 Apr 2024, 00:29
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gmatophobia

Bunuel
­If \(\frac{\sqrt{x^2}}{\sqrt[3]{x^3}}=-1\) and \(x \neq 0\), what is the value of \(\frac{2|x| + 6}{3 - x}\) ?

A. -2
B. 1
C. 2
D. 3
E. Cannot be determined from the given information­
­
\(\frac{\sqrt{x^2}}{\sqrt[3]{x^3}}\)

= \(\frac{|x|}{x}\)

\(\frac{|x|}{x} = -1\)

Hence, we can infer that that \(x\) is negative. 
­
Let's assume \(x = -1\)

= \(\frac{2|x| + 6}{3 - x}\)

= \(\frac{2|-1| + 6}{3 - (-1)}\)

= \(\frac{8}{4}\)

= \(2\)

­Option C
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­I have a doubt. The numerator can be either postive or negative for every case of denominator can only be positive or negative.
In that case, won't the equation be always either 1 or -1. Instead of only -1. This leads to an undefined behaviour right?

Bunuel Can you please help?
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If (x^2)^(1/2)/(x^3)^(1/3) = -1and x 0, what is the value of 27 Apr 2024, 00:46
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OokandGluk
­I have a doubt. The numerator can be either postive or negative for every case of denominator can only be positive or negative.
In that case, won't the equation be always either 1 or -1. Instead of only -1. This leads to an undefined behaviour right?

Bunuel Can you please help?
Show more
­The value of the numerator can only be positive.

For example: 

\(\sqrt{4} = 2\) and not \(\pm2\)­
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If (x^2)^(1/2)/(x^3)^(1/3) = -1and x 0, what is the value of 27 Apr 2024, 01:29
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gmatophobia

Bunuel
­If \(\frac{\sqrt{x^2}}{\sqrt[3]{x^3}}=-1\) and \(x \neq 0\), what is the value of \(\frac{2|x| + 6}{3 - x}\) ?

A. -2
B. 1
C. 2
D. 3
E. Cannot be determined from the given information­
­
\(\frac{\sqrt{x^2}}{\sqrt[3]{x^3}}\)

= \(\frac{|x|}{x}\)

\(\frac{|x|}{x} = -1\)

Hence, we can infer that that \(x\) is negative. 
­
Let's assume \(x = -1\)

= \(\frac{2|x| + 6}{3 - x}\)

= \(\frac{2|-1| + 6}{3 - (-1)}\)

= \(\frac{8}{4}\)

= \(2\)

­Option C
­I have a doubt. The numerator can be either postive or negative for every case of denominator can only be positive or negative.
In that case, won't the equation be always either 1 or -1. Instead of only -1. This leads to an undefined behaviour right?

Bunuel Can you please help?
Show more
\(­\sqrt{x^2} = |x|\) and |x| cannot be negative. Generally, the square root sign always indicates non-negative square root.

Mathematically, \(\sqrt{...}\) 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 (and generally in math) 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:

√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 =-√9 =-3 or x = √9 = 3.

Hope it helps.
 ­
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If (x^2)^(1/2)/(x^3)^(1/3) = -1and x 0, what is the value of 28 Apr 2024, 21:08
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Bunuel
­If \(\frac{\sqrt{x^2}}{\sqrt[3]{x^3}}=-1\) and \(x \neq 0\), what is the value of \(\frac{2|x| + 6}{3 - x}\) ?

A. -2
B. 1
C. 2
D. 3
E. Cannot be determined from the given information­


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If (x^2)^(1/2)/(x^3)^(1/3) = -1and x 0, what is the value of 05 Sep 2025, 05:04
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