Mister Exam
Expression simplification/
Least common denominator/
(z^5-5*z^3+10*z-1/10*z+1/5*z^3-1/z^5)*(z^3-3*z+1/3*z-1/z^3)
Least common denominator (z^5-5*z^3+10*z-1/10*z+1/5*z^3-1/z^5)*(z^3-3*z+1/3*z-1/z^3)
The solution
You have entered
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/ 3 \ | 5 3 z z 1 | / 3 z 1 \ |z - 5*z + 10*z - -- + -- - --|*|z - 3*z + - - --| | 10 5 5| | 3 3| \ z / \ z /
$$\left(\left(\frac{z}{3} + \left(z^{3} - 3 z\right)\right) - \frac{1}{z^{3}}\right) \left(\left(\frac{z^{3}}{5} + \left(- \frac{z}{10} + \left(10 z + \left(z^{5} - 5 z^{3}\right)\right)\right)\right) - \frac{1}{z^{5}}\right)$$
(z^5 - 5*z^3 + 10*z - z/10 + z^3/5 - 1/z^5)*(z^3 - 3*z + z/3 - 1/z^3)
Fraction decomposition
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24/5 + z^(-8) + z^8 - 137*z^2/5 - 112*z^6/15 - 109/(10*z^2) + 8/(3*z^4) + 227*z^4/10
$$z^{8} - \frac{112 z^{6}}{15} + \frac{227 z^{4}}{10} - \frac{137 z^{2}}{5} + \frac{24}{5} - \frac{109}{10 z^{2}} + \frac{8}{3 z^{4}} + \frac{1}{z^{8}}$$
2 6 4
24 1 8 137*z 112*z 109 8 227*z
-- + -- + z - ------ - ------ - ----- + ---- + ------
5 8 5 15 2 4 10
z 10*z 3*z
General simplification
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/ 6 / 2 4\\ / 4 / 2\\
\-10 + z *\99 - 48*z + 10*z //*\-3 + z *\-8 + 3*z //
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8
30*z
$$\frac{\left(z^{4} \left(3 z^{2} - 8\right) - 3\right) \left(z^{6} \left(10 z^{4} - 48 z^{2} + 99\right) - 10\right)}{30 z^{8}}$$
(-10 + z^6*(99 - 48*z^2 + 10*z^4))*(-3 + z^4*(-8 + 3*z^2))/(30*z^8)
Powers
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/ 3 \ / 3 1 8*z\ | 5 1 24*z 99*z| |z - -- - ---|*|z - -- - ----- + ----| | 3 3 | | 5 5 10 | \ z / \ z /
$$\left(z^{3} - \frac{8 z}{3} - \frac{1}{z^{3}}\right) \left(z^{5} - \frac{24 z^{3}}{5} + \frac{99 z}{10} - \frac{1}{z^{5}}\right)$$
(z^3 - 1/z^3 - 8*z/3)*(z^5 - 1/z^5 - 24*z^3/5 + 99*z/10)
Combining rational expressions
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/ 6 / 2 2 / 2\\\ / 4 / 2\\
\-10 + z *\99 + 2*z + 10*z *\-5 + z ///*\-3 + z *\-8 + 3*z //
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8
30*z
$$\frac{\left(z^{4} \left(3 z^{2} - 8\right) - 3\right) \left(z^{6} \left(10 z^{2} \left(z^{2} - 5\right) + 2 z^{2} + 99\right) - 10\right)}{30 z^{8}}$$
(-10 + z^6*(99 + 2*z^2 + 10*z^2*(-5 + z^2)))*(-3 + z^4*(-8 + 3*z^2))/(30*z^8)
Rational denominator
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/ 5 / 3 5 \\ / 3 / 3\\
\-50 + z *\- 240*z + 50*z + 495*z//*\-3 + z *\-8*z + 3*z //
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8
150*z
$$\frac{\left(z^{3} \left(3 z^{3} - 8 z\right) - 3\right) \left(z^{5} \left(50 z^{5} - 240 z^{3} + 495 z\right) - 50\right)}{150 z^{8}}$$
(-50 + z^5*(-240*z^3 + 50*z^5 + 495*z))*(-3 + z^3*(-8*z + 3*z^3))/(150*z^8)
Common denominator
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2 6 4 4 6
24 8 137*z 112*z 227*z -30 - 80*z + 327*z
-- + z - ------ - ------ + ------ - --------------------
5 5 15 10 8
30*z
$$z^{8} - \frac{112 z^{6}}{15} + \frac{227 z^{4}}{10} - \frac{137 z^{2}}{5} + \frac{24}{5} - \frac{327 z^{6} - 80 z^{4} - 30}{30 z^{8}}$$
24/5 + z^8 - 137*z^2/5 - 112*z^6/15 + 227*z^4/10 - (-30 - 80*z^4 + 327*z^6)/(30*z^8)
Numerical answer
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(z^3 - 1/z^3 - 2.66666666666667*z)*(z^5 - 1/z^5 + 9.9*z - 4.8*z^3)
(z^3 - 1/z^3 - 2.66666666666667*z)*(z^5 - 1/z^5 + 9.9*z - 4.8*z^3)
Combinatorics
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/ 4 6\ / 8 10 6\
\-3 - 8*z + 3*z /*\-10 - 48*z + 10*z + 99*z /
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8
30*z
$$\frac{\left(3 z^{6} - 8 z^{4} - 3\right) \left(10 z^{10} - 48 z^{8} + 99 z^{6} - 10\right)}{30 z^{8}}$$
(-3 - 8*z^4 + 3*z^6)*(-10 - 48*z^8 + 10*z^10 + 99*z^6)/(30*z^8)
Trigonometric part
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/ 3 \ / 3 1 8*z\ | 5 1 24*z 99*z| |z - -- - ---|*|z - -- - ----- + ----| | 3 3 | | 5 5 10 | \ z / \ z /
$$\left(z^{3} - \frac{8 z}{3} - \frac{1}{z^{3}}\right) \left(z^{5} - \frac{24 z^{3}}{5} + \frac{99 z}{10} - \frac{1}{z^{5}}\right)$$
(z^3 - 1/z^3 - 8*z/3)*(z^5 - 1/z^5 - 24*z^3/5 + 99*z/10)
Assemble expression
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/ 3 \ / 3 1 8*z\ | 5 1 24*z 99*z| |z - -- - ---|*|z - -- - ----- + ----| | 3 3 | | 5 5 10 | \ z / \ z /
$$\left(z^{3} - \frac{8 z}{3} - \frac{1}{z^{3}}\right) \left(z^{5} - \frac{24 z^{3}}{5} + \frac{99 z}{10} - \frac{1}{z^{5}}\right)$$
(z^3 - 1/z^3 - 8*z/3)*(z^5 - 1/z^5 - 24*z^3/5 + 99*z/10)