(1+x)^2/x^6 doesn't simplify?

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In summary, the equation states that if you remove the top square of an equation, then you can simplify it by reducing the bottom to t^3. However, if you do not have a square root to simplify with, then you can simplify the equation by removing the ^3.
  • #1
Pengwuino
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Ok I got an equation here...
[tex]2\sqrt {\frac{{(1 + 2t^4 )^2 }}{{t^6 }}}[/tex]
is the equation (sorry for misleading title but its the same concept).
Now I was always under the assumption that the top square could be removed as long as you reduce the bottom to [tex]t^3[/tex] since [tex]t^3[/tex] squared is [tex]t^6[/tex] and that you can cancel out squares like that. I guess I'm wrong? I'm running some examples in my mind and I'm kinda realizing you can't do it... but i feel like I've always thought it was true for soem reason.
I also did a simplification through mathematica and found out that the square root actually allows you to remove the [tex]^3[/tex] to get to the [tex]t^3[/tex].
The first question I'm asking is: without that square root... would I be able to simplify?
The 2nd question is exactly why I am able to use that square root to simplify. Sorry if its confusing...
 
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  • #2
Well you can get rid off the square root, since

[tex]
\sqrt {\frac{{a^2 }}
{{b^2 }}} = \sqrt {\left( {\frac{a}
{b}} \right)^2 } = \left| {\frac{a}
{b}} \right|[/tex]
 
  • #3
Yah I was under the impression that even without that square root, it could turn into a/b
 
  • #4
Just remember that the quantity has to remain positive. So the simplification of the original equation is:
[tex]
|{1/t^3 + 2t}|[/tex]
 
  • #5
If you want to see the reduction broken down then this is what you should do.
[tex] 2\sqrt{\frac{(1+t^4)^2}{t^6}} =2\left(\frac{(1+t^4)^{2\frac{1}{2}}}{t^6}\right)^{\frac{1}{2}}[/tex]
[tex]=2\frac{(1+t^4)^{2\frac{1}{2}}}{t^{6\frac{1}{2}}}=2\frac{1+t^4}{t^3} [/tex]
[tex]=2t^{-3}+2t^{-3+4}=|2(t^{-3}+t)| [/tex]

which almost yields the same result as knavish. I think he is missing a factor of 2 somewhere in there as

p.s. sorry I didn't break it down with the absolute value signs, but those were already explained pretty well... and for some reason I like to add them afterwards. bad habit i guess
 
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What is the expression "(1+x)^2/x^6" and why doesn't it simplify?

The expression "(1+x)^2/x^6" is a mathematical expression that involves addition, multiplication, and exponents. It represents a polynomial with a degree of 2 in the numerator and a degree of 6 in the denominator. This expression does not simplify because the terms in the numerator and denominator cannot be combined or cancelled out.

Why is the expression "(1+x)^2/x^6" considered unsimplified?

The expression "(1+x)^2/x^6" is considered unsimplified because it can be written in a simpler form without changing its value. In this case, the expression can be rewritten as (1+x)^2 * x^-6, which is considered a simplified form.

Can the expression "(1+x)^2/x^6" be simplified further?

No, the expression "(1+x)^2/x^6" cannot be simplified further. This is because the terms in the numerator and denominator are already in their simplest form and cannot be combined or reduced any further.

Is it necessary to simplify the expression "(1+x)^2/x^6"?

It depends on the context and purpose of the expression. If the expression is part of a larger problem or equation, simplifying it may make the problem easier to solve. However, if the expression stands alone and does not need to be evaluated, it is not necessary to simplify it.

What is the benefit of simplifying expressions like "(1+x)^2/x^6"?

Simplifying an expression like "(1+x)^2/x^6" can make it easier to understand and work with. It can also help to identify patterns and relationships between different terms in the expression. Additionally, simplifying can make it easier to solve equations involving the expression or use it in other mathematical operations.

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