Mister Exam
Expression simplification/
Least common denominator/
(z^5-5*z^3+10*z-10/z+1/5*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+1/5*z^3-1/z^5)*(z^3-3*z+3/z-1/z^3)
The solution
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/ 3 \ | 5 3 10 z 1 | / 3 3 1 \ |z - 5*z + 10*z - -- + -- - --|*|z - 3*z + - - --| | z 5 5| | z 3| \ z / \ z /
$$\left(\left(\frac{z^{3}}{5} + \left(\left(10 z + \left(z^{5} - 5 z^{3}\right)\right) - \frac{10}{z}\right)\right) - \frac{1}{z^{5}}\right) \left(\left(\left(z^{3} - 3 z\right) + \frac{3}{z}\right) - \frac{1}{z^{3}}\right)$$
(z^5 - 5*z^3 + 10*z - 10/z + z^3/5 - 1/z^5)*(z^3 - 3*z + 3/z - 1/z^3)
General simplification
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/ 2 4 / 2\\ / 4 6 / 2 4\\
-\-1 + 3*z + z *\-3 + z //*\5 + 50*z - z *\50 - 24*z + 5*z //
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8
5*z
$$- \frac{\left(z^{4} \left(z^{2} - 3\right) + 3 z^{2} - 1\right) \left(- z^{6} \left(5 z^{4} - 24 z^{2} + 50\right) + 50 z^{4} + 5\right)}{5 z^{8}}$$
-(-1 + 3*z^2 + z^4*(-3 + z^2))*(5 + 50*z^4 - z^6*(50 - 24*z^2 + 5*z^4))/(5*z^8)
Fraction decomposition
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324/5 + z^(-8) + z^8 - 41/z^2 - 3/z^6 + 13/z^4 - 277*z^2/5 - 39*z^6/5 + 137*z^4/5
$$z^{8} - \frac{39 z^{6}}{5} + \frac{137 z^{4}}{5} - \frac{277 z^{2}}{5} + \frac{324}{5} - \frac{41}{z^{2}} + \frac{13}{z^{4}} - \frac{3}{z^{6}} + \frac{1}{z^{8}}$$
2 6 4
324 1 8 41 3 13 277*z 39*z 137*z
--- + -- + z - -- - -- + -- - ------ - ----- + ------
5 8 2 6 4 5 5 5
z z z z
Combinatorics
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3 3 / 4 8 10 6\
(1 + z) *(-1 + z) *\-5 - 50*z - 24*z + 5*z + 50*z /
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8
5*z
$$\frac{\left(z - 1\right)^{3} \left(z + 1\right)^{3} \left(5 z^{10} - 24 z^{8} + 50 z^{6} - 50 z^{4} - 5\right)}{5 z^{8}}$$
(1 + z)^3*(-1 + z)^3*(-5 - 50*z^4 - 24*z^8 + 5*z^10 + 50*z^6)/(5*z^8)
Combining rational expressions
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/ 2 / 2 / 2\\\ / 4 / 4 2 / 2 / 2\\\\
\-1 + z *\3 + z *\-3 + z ///*\-5 + z *\-50 + z + 5*z *\10 + z *\-5 + z ////
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8
5*z
$$\frac{\left(z^{2} \left(z^{2} \left(z^{2} - 3\right) + 3\right) - 1\right) \left(z^{4} \left(z^{4} + 5 z^{2} \left(z^{2} \left(z^{2} - 5\right) + 10\right) - 50\right) - 5\right)}{5 z^{8}}$$
(-1 + z^2*(3 + z^2*(-3 + z^2)))*(-5 + z^4*(-50 + z^4 + 5*z^2*(10 + z^2*(-5 + z^2))))/(5*z^8)
Assemble expression
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/ 3\ / 3 1 3\ | 5 1 10 24*z | |z - -- - 3*z + -|*|z - -- - -- + 10*z - -----| | 3 z| | 5 z 5 | \ z / \ z /
$$\left(z^{3} - 3 z + \frac{3}{z} - \frac{1}{z^{3}}\right) \left(z^{5} - \frac{24 z^{3}}{5} + 10 z - \frac{10}{z} - \frac{1}{z^{5}}\right)$$
(z^3 - 1/z^3 - 3*z + 3/z)*(z^5 - 1/z^5 - 10/z + 10*z - 24*z^3/5)
Rational denominator
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/ 3 / / 3 \\\ / 5 / 4 / 5 3 \\\
\-z + z *\3 + z*\z - 3*z///*\-5*z + z *\-50 + z + 5*z*\z - 5*z + 10*z///
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10
5*z
$$\frac{\left(z^{3} \left(z \left(z^{3} - 3 z\right) + 3\right) - z\right) \left(z^{5} \left(z^{4} + 5 z \left(z^{5} - 5 z^{3} + 10 z\right) - 50\right) - 5 z\right)}{5 z^{10}}$$
(-z + z^3*(3 + z*(z^3 - 3*z)))*(-5*z + z^5*(-50 + z^4 + 5*z*(z^5 - 5*z^3 + 10*z)))/(5*z^10)
Trigonometric part
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/ 3\ / 3 1 3\ | 5 1 10 24*z | |z - -- - 3*z + -|*|z - -- - -- + 10*z - -----| | 3 z| | 5 z 5 | \ z / \ z /
$$\left(z^{3} - 3 z + \frac{3}{z} - \frac{1}{z^{3}}\right) \left(z^{5} - \frac{24 z^{3}}{5} + 10 z - \frac{10}{z} - \frac{1}{z^{5}}\right)$$
(z^3 - 1/z^3 - 3*z + 3/z)*(z^5 - 1/z^5 - 10/z + 10*z - 24*z^3/5)
Numerical answer
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(z^3 - 1/z^3 + 3.0/z - 3.0*z)*(z^5 - 1/z^5 + 10.0*z - 10.0/z - 4.8*z^3)
(z^3 - 1/z^3 + 3.0/z - 3.0*z)*(z^5 - 1/z^5 + 10.0*z - 10.0/z - 4.8*z^3)
Common denominator
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2 6 4 4 2 6
324 8 277*z 39*z 137*z -1 - 13*z + 3*z + 41*z
--- + z - ------ - ----- + ------ - -------------------------
5 5 5 5 8
z
$$z^{8} - \frac{39 z^{6}}{5} + \frac{137 z^{4}}{5} - \frac{277 z^{2}}{5} + \frac{324}{5} - \frac{41 z^{6} - 13 z^{4} + 3 z^{2} - 1}{z^{8}}$$
324/5 + z^8 - 277*z^2/5 - 39*z^6/5 + 137*z^4/5 - (-1 - 13*z^4 + 3*z^2 + 41*z^6)/z^8
Powers
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/ 3\ / 3 1 3\ | 5 1 10 24*z | |z - -- - 3*z + -|*|z - -- - -- + 10*z - -----| | 3 z| | 5 z 5 | \ z / \ z /
$$\left(z^{3} - 3 z + \frac{3}{z} - \frac{1}{z^{3}}\right) \left(z^{5} - \frac{24 z^{3}}{5} + 10 z - \frac{10}{z} - \frac{1}{z^{5}}\right)$$
(z^3 - 1/z^3 - 3*z + 3/z)*(z^5 - 1/z^5 - 10/z + 10*z - 24*z^3/5)