sologuitar Posted December 4, 2022 Share Posted December 4, 2022 I am stuck on a question concerning a square root prove. Here it is: Give an example to show that sqrt{a^(2)} is not equal to a. Use it to explain why sqrt{a^(2)} = | a |. Link to comment Share on other sites More sharing options...

Endy0816 Posted December 4, 2022 Share Posted December 4, 2022 Negative numbers. 1 Link to comment Share on other sites More sharing options...

Country Boy Posted December 16, 2022 Share Posted December 16, 2022 a= -2. a^{2}= 4. What is the square root of 4? (Remember that the square root is a FUNCTION so can have only one value.) 1 Link to comment Share on other sites More sharing options...

Wanderlust12 Posted February 15 Share Posted February 15 By convention, we refer to the major square root function when you say "THE square root." When working with real numbers, the primary square root always yields a non-negative value. Consider the two negative numbers multiplied together become a positive number, so x * x = x ^ 2 but, also, -x * -x = x^2 In most generalized questions the positive answer (x) is given, but technically -x is also an answer 1 Link to comment Share on other sites More sharing options...

Country Boy Posted March 19 Share Posted March 19 (edited) No! Especially since you used the word "technically". Technically, [math]\sqrt{x^2}= |x|[/math]. That is x if [math]x\ge 0[/math] and -x if [math]x< 0[/math], Edited March 19 by Country Boy 1 Link to comment Share on other sites More sharing options...

Genady Posted March 19 Share Posted March 19 28 minutes ago, Country Boy said: No! Especially since you used the word "technically". Technically, x2−−√=|x| . That is x if x≥0 and -x if x<0 , AFAIK, x^{2} has two square roots: +x and -x. 1 Link to comment Share on other sites More sharing options...

Country Boy Posted March 25 Share Posted March 25 On 3/19/2023 at 6:50 PM, Genady said: AFAIK, x^{2} has two square roots: +x and -x. For a given positive number, a, both x=√a and -x= -√a satisfy the equation [math]x^2= a[/math] and are called "square roots of a". The square root function, √x, however, in order to be a function, must have a single value and that is defined to be the positive value. "The" square root of x^2 is |x|. 1 Link to comment Share on other sites More sharing options...

Genady Posted March 25 Share Posted March 25 6 minutes ago, Country Boy said: For a given positive number, a, both x=√a and -x= -√a satisfy the equation x2=a and are called "square roots of a". The square root function, √x, however, in order to be a function, must have a single value and that is defined to be the positive value. "The" square root of x^2 is |x|. OK. I didn't know the OP is about the function since it didn't say so. Link to comment Share on other sites More sharing options...

studiot Posted March 25 Share Posted March 25 3 hours ago, Genady said: OK. I didn't know the OP is about the function since it didn't say so. Actually the OP described a function since it used a letter for a variable, not a specific number. It is important to distinguish between a function and the value of that function at a specific point. A square root of a number is another number, a square root of a function is another function. So a square root of 4 is 2 and another square root is -2, all of which are numbers. Folks also often make this mistake with the differential calculus, where the derived function (the derivative) has, possibly different, values at every point where the original function is differentiable. 1 Link to comment Share on other sites More sharing options...

NTuft Posted March 28 Share Posted March 28 On 12/3/2022 at 7:18 PM, sologuitar said: I am stuck on a question concerning a square root prove. Here it is: Give an example to show that sqrt{a^(2)} is not equal to a. Use it to explain why sqrt{a^(2)} = | a |. On 12/16/2022 at 6:46 AM, Country Boy said: a= -2. a^{2}= 4. What is the square root of 4? (Remember that the square root is a FUNCTION so can have only one value.) The square root may be considered a multivalued function. On 3/19/2023 at 4:50 PM, Genady said: AFAIK, x^{2} has two square roots: +x and -x. Example 1: a=-2 a^{2}=+4 Example 2: a=+2 a^{2}=4 The square operation should be applied first once a number is input to evaluate the expression. In distinction from the algebra convention when solving for variables where the power is simply removed. Squaring of course gives a positive value. Once the square root is applied, plus/minus is appended in front of the operator, for the equivalent reason absolute value is used here. ; because there is ambiguity (multi-value) output from the square root... relation. Nothing in the question remarked about limiting the domain of input or codomain of output to ensure a single-valued function. 1 Link to comment Share on other sites More sharing options...

MartaSher Posted April 9 Share Posted April 9 Very interesting information. 1 Link to comment Share on other sites More sharing options...

JermaineKim Posted April 25 Share Posted April 25 On 12/4/2022 at 9:18 AM, sologuitar said: I am stuck on a question concerning a square root prove. Here it is: Give an example to show that sqrt{a^(2)} is not equal to a. Use it to explain why sqrt{a^(2)} = | a |. First, let's take an example to show that sqrt{a^(2)} is not equal to a. Suppose a = -3, then: sqrt{a^(2)} = sqrt{(-3)^(2)} = sqrt{9} = 3 But a = -3, so sqrt{a^(2)} is not equal to a. Now, let's move on to explain why sqrt{a^(2)} = | a |. The square root of a number is always a positive number or zero. Therefore, when we take the square root of a square, we need to make sure that the result is also positive. In the example above, we took the square root of (-3)^2, which gave us 3. We know that the square of a number is always positive, so (-3)^2 is equal to 9. However, since a can be negative or positive, we need to use the absolute value of a to make sure that we get a positive result when we take the square root. Thus, we can conclude that sqrt{a^(2)} = | a |. 1 Link to comment Share on other sites More sharing options...

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