Understanding Simplification Operations for Inverse Trigonometric Functions

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The discussion focuses on simplifying the expression involving inverse trigonometric functions and the origin of the term (1 - 4t^6) in the denominator. The user struggles to understand how this term arises when simplifying the square root in the denominator of a complex fraction. By factoring out 1/4t^6 from the expression, the relationship between the terms becomes clearer. The simplification process reveals that (1 - 4t^6) is derived from the difference between (1/4t^6) and 1. The conversation highlights common challenges in calculus related to simplification and derivative calculations.
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This is a question on the simplification operations. I can't for the life of me figure out how:

\frac{1}{\frac{1}{2t^3}\sqrt{(\frac{1}{2t^3})^2 -1}}*(-\frac{3}{2t^4})= -\frac{3}{t\sqrt{\frac{1}{4t^6}(1 - 4t^6)}}

Really, I can't figure out where (1-4t^6) is coming from!

It's involved in finding the derivative of inverse trigonometric function and I'm getting stuck right in the middle with the easy stuff.
 
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\left( \frac{1}{2t^3} \right)^2 -1 = \frac{1-4t^6}{4t^6} just by pulling a factor of \frac{1}{4t^6} out to the front.
 
Well, note that
\left(\frac{1}{2t^3} \right)^2 = \frac{1}{4t^6}

So, looking inside that square root in the denominator...
\begin{aligned}<br /> \left(\frac{1}{2t^3} \right)^2 - 1 &amp;= \frac{1}{4t^6} - 1 \\<br /> &amp;= \frac{1}{4t^6} - \frac{4t^6}{4t^6} \\<br /> &amp;= \frac{1}{4t^6}(1) - \frac{1}{4t^6}(4t^6) \\<br /> &amp;= \frac{1}{4t^6}(1 - 4t^6)<br /> \end{aligned}

EDIT: Beaten to it. :wink:
 
Doh! Thanks guys :P
 
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Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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