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I've been studying for the GMAT, and I have a question regarding the use of absolute values under a square root in certain mathematical expressions. I hope someone can provide some clarity.
In some practice questions, I've encountered expressions like √(|x|) or √(|a - b|). My question is: when do I need to consider the absolute value, and when can I directly simplify expressions without it?
Scenario 1: If I have √(x^2), can I simplify this to |x| or just x? When is it necessary to use the absolute value?
Scenario 2: When dealing with expressions like √(|a - b|), what are the rules for simplifying this type of expression, credit cardholders and when should I retain the absolute value?
I want to make sure I have a clear understanding of when to use absolute values in such situations to tackle GMAT quant questions effectively. Any insights or examples would be greatly appreciated. Thank you in advance for your help!
In some practice questions, I've encountered expressions like √(|x|) or √(|a - b|). My question is: when do I need to consider the absolute value, and when can I directly simplify expressions without it?
Scenario 1: If I have √(x^2), can I simplify this to |x| or just x? When is it necessary to use the absolute value?
Scenario 2: When dealing with expressions like √(|a - b|), what are the rules for simplifying this type of expression, credit cardholders and when should I retain the absolute value?
I want to make sure I have a clear understanding of when to use absolute values in such situations to tackle GMAT quant questions effectively. Any insights or examples would be greatly appreciated. Thank you in advance for your help!
06 Sep 2023, 06:04
Kudos
Bookmarks
Ana_Grace
Great questions! Let's break down each of your scenarios:
Scenario 1: \(\sqrt{x^2}\)
The expression \(\sqrt{x^2}\) is indeed equal to |x|, not just x. This is because the square root function always returns non-negative values, and |x| ensures that the result is non-negative for all x.
For instance, consider x = -5:
\(\sqrt{(-5)^2} = \sqrt{25} = 5\)
However, if we just took x directly, we'd get -5, which is not correct.
Thus, \(\sqrt{x^2} = |x|\).
Scenario 2: \(\sqrt{|a - b|}\)
The absolute value inside the square root ensures that the value under the root is non-negative, because the square root of a negative number is not a real number and thus not defined on the GMAT. So, this expression ensures we're working with real numbers.
To simplify \(\sqrt{|a - b|}\) (or determine when you can drop the absolute value), you'd typically need some context about the relationship between a and b.
- 1. If you know that \(a \geq b\):
Then a - b is non-negative, and you don't need the absolute value. In this case, \(\sqrt{|a - b|} = \sqrt{a - b}\).
2. If you know that \(a < b\):
Then a - b is negative, and the absolute value is necessary to make it positive. In this case, \(\sqrt{|a - b|} = \sqrt{b - a}\).
3. If you don't have any information about the relationship between a and b:
Then you cannot drop the absolute value, as you don't know the sign of a - b.
Practical Takeaways for the GMAT:
1. Always remember that the square root of a number squared is its absolute value: \(\sqrt{x^2} = |x|\).
2. When you see absolute values inside square roots, try to identify the context or constraints given in the problem. If you know the order of the numbers, you can often simplify the expression further.
3. In many GMAT problems, being able to recognize and simplify such expressions quickly can be the key to answering a question correctly under time pressure.
Hope it helps.
