The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- a^{4} - 8 a^{2}\right) - 1$$
To do this, let's use the formula
$$a^{5} + a^{2} b + c = a \left(a^{2} + m\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = -1$$
$$b = -8$$
$$c = -1$$
Then
$$m = 4$$
$$n = 15$$
So,
$$-10$$
General simplification
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$$- a^{4} - 8 a^{2} - 1$$
/ ____________\ / ____________\ / ____________\ / ____________\
| / ____ | | / ____ | | / ____ | | / ____ |
\a + I*\/ 4 - \/ 15 /*\a - I*\/ 4 - \/ 15 /*\a + I*\/ 4 + \/ 15 /*\a - I*\/ 4 + \/ 15 /
$$\left(a - i \sqrt{4 - \sqrt{15}}\right) \left(a + i \sqrt{4 - \sqrt{15}}\right) \left(a + i \sqrt{\sqrt{15} + 4}\right) \left(a - i \sqrt{\sqrt{15} + 4}\right)$$
(((a + i*sqrt(4 - sqrt(15)))*(a - i*sqrt(4 - sqrt(15))))*(a + i*sqrt(4 + sqrt(15))))*(a - i*sqrt(4 + sqrt(15)))
Combining rational expressions
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$$a^{2} \left(- a^{2} - 8\right) - 1$$
$$- a^{4} - 8 a^{2} - 1$$
$$- a^{4} - 8 a^{2} - 1$$
$$- a^{4} - 8 a^{2} - 1$$
Assemble expression
[src]
$$- a^{4} - 8 a^{2} - 1$$
Rational denominator
[src]
$$- a^{4} - 8 a^{2} - 1$$
$$- a^{4} - 8 a^{2} - 1$$