Simple math rules seem contradictory

[Holocene]

Simple math rules seem contradictory... :(

Consider this simple expression:

$$x^8 - 16$$

If we wanted to write this expression as the product of two factors, we could start with something simple like this:

$$\sqrt{x^8 -16}$$ . $$\sqrt{x^8 -16}$$

From that, we would simply get this:

$$(x^4 - 4)(x^4 - 4)$$

This is wrong though, as it does not equal the original expression. Multiplying two negative values will result in a positive value for 16. This is false, as the original exprsssion clearly has a negative value for 16.

So, one of the signs in $$(x^4 - 4)(x^4 - 4)$$ must chnage to a possitive sign.

It just seem to me like, at times, some of the mathematical rules seem contradictory.

Or am I wrong in that you cannot take the individual roots of the terms in an expression?

Anyone information would be greatly appreciated?

 

[waht]

Actually,

$$x^8 -16 = (x^4 +4)(x^4 -4)$$

What is the point of breaking the expression into radicals?

 

[D H]

Waht is exactly right. The only problem with his/her answer is that it wasn't explicit enough.

Or am I wrong in that you cannot take the individual roots of the terms in an expression?

Holocene, what you did in the original post was wrong. The square root does not distribute over subtraction. You cannot take individual roots of the terms in an expression. Example: Consider the Pythagorean triple $3^2+4^2=5^2$, so $\sqrt{5^2-4^2} = 3$. Taking roots of individual terms as was done in the OP would lead to 5-4=1, obviously not 3.

 

[JasonRox]

Holocene, what you did in the original post was wrong. The square root does not distribute over subtraction. You cannot take individual roots of the terms in an expression. Example: Consider the Pythagorean triple $3^2+4^2=5^2$, so $\sqrt{5^2-4^2} = 3$. Taking roots of individual terms as was done in the OP would lead to 5-4=1, obviously not 3.

So many students make this mistake!

 

[Invictious]

Consider this simple expression:

$$x^8 - 16$$

$$\sqrt{x^8 -16}$$ . $$\sqrt{x^8 -16}$$

$$(x^4 - 4)(x^4 - 4)$$

This is wrong though, as it does not equal the original expression. Multiplying two negative values will result in a positive value for 16. This is false, as the original exprsssion clearly has a negative value for 16.

So, one of the signs in $$(x^4 - 4)(x^4 - 4)$$ must chnage to a possitive sign.

It just seem to me like, at times, some of the mathematical rules seem contradictory.

Or am I wrong in that you cannot take the individual roots of the terms in an expression?

Anyone information would be greatly appreciated?

It should be the difference of squares, so it should have (a+b)(a-b).
Your breaking apart of the expression into radical was wrong. So no, the mathematical rules do not contradict each other.

 

[arildno]

The correct radicand decomposition is:
$$x^{8}-16=sign(x^{8}-16)\sqrt{|x^{8}-16|}\sqrt{|x^{8}-16|}$$

 

[ice109]

wow no wonder i couldn't figure out how he reduced the roots