Math ACT question

Test Preparation ACT Preparation
1

<p>Question: </p>

<p>Which of the following expressions, if any, are equal for all real numbers x?</p>

<p>I. √(-x)^2 (the square is under the square root)</p>

<p>II. |-x|</p>

<p>III. -|x|</p>

<p>A. I and II only
B. I and III only
C. II and III only
D. I, II, and III
E. None of the expressions are equivalent</p>

<p>Book answer: A</p>

<p>In the red book, there is a question that claims √(-x)^2 (the square is under the square root) and |-x| are equal. This does not make sense to me. It is clear that |-x| has to be positive, because absolute value is the offset from zero, and therefore must be positive. But I do not understand how the first expression has to be positive. </p>

<p>Lets say that x = 3. There are 2 ways that I know to evaluate this. </p>

<li><p>if x = 3, then -x = -3, and -3^2 is 9. the √9 = 3 and -3. </p></li>
<li><p>if x = 3, then -x = -3. because ^2 = ^(2/1) and √ = ^(1/2), then -3^((2/1)(1/2))
= -3^(2/2) = -3^(1) = -3. </p></li>
</ol>

<p>Lets say that x = -3 (remember that when x = -3, -x = 3).</p>

<li><p>Using the same method shown when x = 3 (now being used for x = -3), 3^2 = 9,
√9 = 3 and -3.</p></li>
<li><p>Using the same method shown when x = 3 (now being used for x = -3), 3^1 = 3.</p></li>
</ol>

<p>This last solution is the only one that works, but 3 is a real number and |3| = 3. If II has to be positive for a number x (a real number), and I = II, then I must equal II (for the same number x). Yet when 3 = x, I (think I) proved that II (|-3|) = 3, and I = 3 and -3 (using method 1) or -3 (using method 2). REGARDLESS of which method is more correct, there is still a -3 as an option for the answer of I, and the question asks for the 2 expressions that are equal for all real numbers. </p>

<p>Can someone please prove me wrong? Is my logic just flawed? </p>

<p>Thank you for reading!</p>

2

<p>lol, i just did this practice test today. second one on the red book!</p>

<p>the flaw in your logic comes from the fact that you assume that sqrt(-x^2) can be negative. although it is true that it CAN be negative mathematically, it is common standard to ONLY use the positive root, the principle root. therefore, the I option is positive and II option is also positive. </p>

<p>hope this answers your question</p>

3

<p>thats such bs, the square root of a number is negative and positive. There shouldn’t be a question on a standardized test that is dependent on subjective non-mathematical practices. Arg, that makes me angry.</p>

<p>Thanks for your answer though!</p>