Mister Exam
Expression simplification/
Least common denominator/
(z^5-5*z^3+10*z-10/z+5/z^3-1/z^5)*(z^3-3*z+3/z-1/z^3)
Least common denominator (z^5-5*z^3+10*z-10/z+5/z^3-1/z^5)*(z^3-3*z+3/z-1/z^3)
The solution
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/ 5 3 10 5 1 \ / 3 3 1 \ |z - 5*z + 10*z - -- + -- - --|*|z - 3*z + - - --| | z 3 5| | z 3| \ z z / \ z /
$$\left(\left(\left(z^{3} - 3 z\right) + \frac{3}{z}\right) - \frac{1}{z^{3}}\right) \left(\left(\left(\left(10 z + \left(z^{5} - 5 z^{3}\right)\right) - \frac{10}{z}\right) + \frac{5}{z^{3}}\right) - \frac{1}{z^{5}}\right)$$
(z^5 - 5*z^3 + 10*z - 10/z + 5/z^3 - 1/z^5)*(z^3 - 3*z + 3/z - 1/z^3)
General simplification
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1 8 56 2 8 6 28 4
70 + -- + z - -- - 56*z - -- - 8*z + -- + 28*z
8 2 6 4
z z z z
$$z^{8} - 8 z^{6} + 28 z^{4} - 56 z^{2} + 70 - \frac{56}{z^{2}} + \frac{28}{z^{4}} - \frac{8}{z^{6}} + \frac{1}{z^{8}}$$
70 + z^(-8) + z^8 - 56/z^2 - 56*z^2 - 8/z^6 - 8*z^6 + 28/z^4 + 28*z^4
Fraction decomposition
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70 + z^(-8) + z^8 - 56/z^2 - 56*z^2 - 8/z^6 - 8*z^6 + 28/z^4 + 28*z^4
$$z^{8} - 8 z^{6} + 28 z^{4} - 56 z^{2} + 70 - \frac{56}{z^{2}} + \frac{28}{z^{4}} - \frac{8}{z^{6}} + \frac{1}{z^{8}}$$
1 8 56 2 8 6 28 4
70 + -- + z - -- - 56*z - -- - 8*z + -- + 28*z
8 2 6 4
z z z z
Trigonometric part
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/ 3 1 3\ / 5 1 10 3 5 \ |z - -- - 3*z + -|*|z - -- - -- - 5*z + -- + 10*z| | 3 z| | 5 z 3 | \ z / \ z z /
$$\left(z^{3} - 3 z + \frac{3}{z} - \frac{1}{z^{3}}\right) \left(z^{5} - 5 z^{3} + 10 z - \frac{10}{z} + \frac{5}{z^{3}} - \frac{1}{z^{5}}\right)$$
(z^3 - 1/z^3 - 3*z + 3/z)*(z^5 - 1/z^5 - 10/z - 5*z^3 + 5/z^3 + 10*z)
Combinatorics
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8 8
(1 + z) *(-1 + z)
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8
z
$$\frac{\left(z - 1\right)^{8} \left(z + 1\right)^{8}}{z^{8}}$$
(1 + z)^8*(-1 + z)^8/z^8
Powers
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/ 3 1 3\ / 5 1 10 3 5 \ |z - -- - 3*z + -|*|z - -- - -- - 5*z + -- + 10*z| | 3 z| | 5 z 3 | \ z / \ z z /
$$\left(z^{3} - 3 z + \frac{3}{z} - \frac{1}{z^{3}}\right) \left(z^{5} - 5 z^{3} + 10 z - \frac{10}{z} + \frac{5}{z^{3}} - \frac{1}{z^{5}}\right)$$
(z^3 - 1/z^3 - 3*z + 3/z)*(z^5 - 1/z^5 - 10/z - 5*z^3 + 5/z^3 + 10*z)
Rational denominator
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/ 3 / / 3 \\\ / 4 5 / 3 / / 5 3 \\\\
\-z + z *\3 + z*\z - 3*z///*\- z + z *\5*z + z *\-10 + z*\z - 5*z + 10*z////
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13
z
$$\frac{\left(z^{3} \left(z \left(z^{3} - 3 z\right) + 3\right) - z\right) \left(z^{5} \left(z^{3} \left(z \left(z^{5} - 5 z^{3} + 10 z\right) - 10\right) + 5 z\right) - z^{4}\right)}{z^{13}}$$
(-z + z^3*(3 + z*(z^3 - 3*z)))*(-z^4 + z^5*(5*z + z^3*(-10 + z*(z^5 - 5*z^3 + 10*z))))/z^13
Common denominator
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4 2 6
8 2 6 4 -1 - 28*z + 8*z + 56*z
70 + z - 56*z - 8*z + 28*z - -------------------------
8
z
$$z^{8} - 8 z^{6} + 28 z^{4} - 56 z^{2} + 70 - \frac{56 z^{6} - 28 z^{4} + 8 z^{2} - 1}{z^{8}}$$
70 + z^8 - 56*z^2 - 8*z^6 + 28*z^4 - (-1 - 28*z^4 + 8*z^2 + 56*z^6)/z^8
Combining rational expressions
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/ 2 / 2 / 2\\\ / 2 / 2 / 2 / 2 / 2\\\\\
\-1 + z *\3 + z *\-3 + z ///*\-1 + z *\5 + z *\-10 + z *\10 + z *\-5 + z /////
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8
z
$$\frac{\left(z^{2} \left(z^{2} \left(z^{2} - 3\right) + 3\right) - 1\right) \left(z^{2} \left(z^{2} \left(z^{2} \left(z^{2} \left(z^{2} - 5\right) + 10\right) - 10\right) + 5\right) - 1\right)}{z^{8}}$$
(-1 + z^2*(3 + z^2*(-3 + z^2)))*(-1 + z^2*(5 + z^2*(-10 + z^2*(10 + z^2*(-5 + z^2)))))/z^8
Assemble expression
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/ 3 1 3\ / 5 1 10 3 5 \ |z - -- - 3*z + -|*|z - -- - -- - 5*z + -- + 10*z| | 3 z| | 5 z 3 | \ z / \ z z /
$$\left(z^{3} - 3 z + \frac{3}{z} - \frac{1}{z^{3}}\right) \left(z^{5} - 5 z^{3} + 10 z - \frac{10}{z} + \frac{5}{z^{3}} - \frac{1}{z^{5}}\right)$$
(z^3 - 1/z^3 - 3*z + 3/z)*(z^5 - 1/z^5 - 10/z - 5*z^3 + 5/z^3 + 10*z)
Numerical answer
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(z^3 - 1/z^3 + 3.0/z - 3.0*z)*(z^5 - 1/z^5 + 5.0/z^3 + 10.0*z - 5.0*z^3 - 10.0/z)
(z^3 - 1/z^3 + 3.0/z - 3.0*z)*(z^5 - 1/z^5 + 5.0/z^3 + 10.0*z - 5.0*z^3 - 10.0/z)