Square Root Prove

[sologuitar]

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 |. 

 

 

[Endy0816]

Negative numbers.

[Country Boy]

a= -2.

a²= 4.  

What is the square root of 4?

(Remember that the square root is a FUNCTION so can have only one value.)

[Wanderlust12]

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

[Country Boy]

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, 2023 by Country Boy

[Genady]

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, x² has two square roots: +x and -x.

[Country Boy]

On 3/19/2023 at 6:50 PM, Genady said:

AFAIK, x² 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|.

[Genady]

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.

[studiot]

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.

 

 

[NTuft]

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.

a²= 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, x² has two square roots: +x and -x.

Example 1:
a=-2
a²=+4

Example 2:
a=+2
a²=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. 

[MartaSher]

Very interesting information. 

[JermaineKim]

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 |.