Simple square root factoring question

In summary, the conversation was about whether the expression \sqrt{a^2-a^2\sin^2{x}} is equal to a\cos{x}. It was initially deemed correct, but it was then pointed out that it should be written as \|a\cos x\| to account for the possibility of negative values. There was also a mention of the modulus or absolute value.
  • #1
Odyssey
87
0
Hi,

is [tex]\sqrt{a^2-a^2\sin^2{x}} = a\cos{x}?[/tex]

If not, what should it be? :confused:

Appreciate the help! :smile:
 
Last edited:
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  • #2
That is correct.
 
  • #3
Thank you for the help Sirus!
 
  • #4
It is correct within a sign!
 
  • #5
Tide said:
It is correct within a sign!

What do you mean?
 
  • #6
Sirus said:
That is correct.

No, that is not quite correct if you think of how a secondary definition of the modulus (absolute value) follows from the square root of a square,

[tex]\sqrt{x^2} = \|x\|[/tex]

In your case

[tex]\sqrt{a^2-a^2\sin^2{x}} = a\cos{x}[/tex]

can be written if and only if a and cos(x) are both positive or both negative; before you brought them out of the square root sign, you had the intermediate step,

[tex]\sqrt{a^2-a^2\sin^2{x}} = \sqrt{a^2\cos^2{x}}[/tex]

so

[tex]\sqrt{a^2-a^2\sin^2{x}} = \sqrt{a^2\cos^2{x}} = \|a\cos x\|[/tex]

which should hold true anyway since the left hand side is the positive square root.

In general however, you can write it as a cos x if you have no major problems with the signs (you won't have any if theta lies between 0 and pi/2 and a > 0 for instance). But if you're proving something which involves this substitution, you had rather take it into account.

Cheers
Vivek
 
  • #7
I stand corrected. I forgot about that.
 

What is simple square root factoring?

Simple square root factoring is a process of finding the factors of a number that can be written as the product of two identical numbers. In other words, it is finding the square root of a number.

How do I factor a simple square root?

To factor a simple square root, you need to find the square root of the given number. You can do this by taking the square root of both the numbers inside the radical sign and then simplifying the expression.

What are the steps for factoring a simple square root?

The steps for factoring a simple square root are:
1. Identify the number inside the radical sign
2. Find the square root of that number
3. Simplify the expression using basic algebraic rules
4. Check your answer by multiplying the factors back together

What is the difference between factoring a simple square root and a complex square root?

The main difference between factoring a simple square root and a complex square root is that in a simple square root, the number inside the radical sign is a perfect square, while in a complex square root, the number inside the radical sign is not a perfect square. This means that the square root of a simple square root will be a whole number, while the square root of a complex square root will be a decimal or an irrational number.

Can I factor a negative square root?

No, you cannot factor a negative square root. This is because the square root of a negative number is an imaginary number, which cannot be factored using real numbers.

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