Explain the difference between these square roots

In summary: That is why \sqrt{a} is defined to be the positive root of a, so that \sqrt{a} is a function.In summary, the difference between the two statements V¯(x) = ± 4 and V¯(x) = - 4 is that the first one is incorrect, as the square root symbol only represents the positive square root. The correct notation is \pm \sqrt{x} = \pm 2, where the plus or minus sign accounts for both the positive and negative solutions. This is a reminder from the text that the square root symbol always means the positive root, and should be used correctly in all cases.
  • #1
greenneub
16
0
Hey guys, I was just wondering what the difference between these two statements are:

V¯(x) = ± 4

V¯(x) = - 4 ---> does not exist.

This is the quote from my text, "...we remind you of a very important agreement in mathematics. The square root sign V¯ always means take the positive square root of whatever is under it. For instance, V¯(4) = 2, it is not equal to -2, only 2. Keep this in mind in this section, and always. "

Maybe I've been staring at the pages too long, but how is (4) different from (-2)²? And why can we right ± 2, but not -2? I know this is basic but I'm embarrassingly confused about this.
 
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  • #2
[itex]\sqrt{x}[/itex] means the positive square root of x (this way you can refer to [itex] \sqrt{x}[/itex] and only be talking about a single number, not two numbers). If the author says [itex]\sqrt{16}=\pm 4[/itex], he is just making the point that both [itex]4^2[/itex] and [itex](-4)^2[/itex] equal 16, however the correct notation is[itex]\pm \sqrt{16}=\pm 4[/itex].
 
  • #3
As qntty said, the first, [tex]\sqrt{4}= \pm 2[/itex] is simply wrong. [tex]\sqrt{4}= 2[/tex] because [tex]\sqrt{x}[/tex] is defined as the positive number y such that [tex]y^2= x[/tex]. That is why we must write the solution to [tex]x^2= a[/tex] as [itex]\pm\sqrt{a}[/itex]- because [itex]\sqrt{a}[/itex] does not include "[tex]\pm[/tex]".
 
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  • #4
And also because that we wish that square-root should be a "function", and for being a function it has to be defined like that only. By definition, a function takes a value from a set A and maps it into B, and no two numbers in A can map to the same number in B.
 

1. What is the difference between square roots and cube roots?

Square roots and cube roots are both types of mathematical operations that involve finding the number that, when multiplied by itself a certain number of times, equals a given number. The main difference between the two is that square roots involve finding a number that, when multiplied by itself once, equals a given number, while cube roots involve finding a number that, when multiplied by itself twice, equals a given number.

2. How do you calculate square roots and cube roots?

To calculate a square root, you can use a calculator or a mathematical formula. The formula for finding the square root of a number is to take the number and divide it by 2. Then, keep taking the average of the result and the original number until you reach a close approximation. To calculate a cube root, you can also use a calculator or a formula. The formula for finding the cube root of a number is to take the number and divide it by 3. Then, keep taking the average of the result and the original number until you reach a close approximation.

3. What is the difference between the symbol for square root and cube root?

The symbol for square root is a radical sign (√), while the symbol for cube root is a radical sign with a small 3 on the top (∛). This small 3 indicates that the number is being multiplied by itself three times.

4. Can square roots and cube roots be negative?

Yes, square roots and cube roots can be negative. However, when finding the square root of a negative number, the answer is considered to be an imaginary number. When finding the cube root of a negative number, the answer is a real number, but it may also have a negative value.

5. How are square roots and cube roots used in real life?

Square roots and cube roots have many practical applications in fields such as engineering, physics, and finance. For example, in engineering, square roots and cube roots are used to calculate the dimensions of shapes and structures. In physics, they are used to calculate forces and velocities. In finance, they are used to calculate interest rates and growth rates. They also have applications in everyday life, such as calculating the area of a room or the volume of a container.

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