Lösungen

1

Remove["Global`*"]

x1 = 2.1898989898989;
x10 = 10 x1;
x1000 = 100 x10;
x990 = x1000 - x10 // Round;
y1 = 19.79;
y100 = 1979.79;
y99 = y100 - y1 // Round;
z1 = 18.97;
z100 = 1897.97;
z99 = z100 - z1 // Round;

x = x990/990

N[%, 20]

y = y99/99

N[%, 20]

z = z99/99

N[%, 20]

x/(y - z)

N[%, 20]

Kontrolle:

x1/(y1 - z1)

(2168/990)/(81736/99900) // N

2

Remove["Global`*"]

g1 = (4  y - 2  z + w == 14);
g2 = (y + 2  z + w == 56);
g3 = (3  y - 4  z == 40);
Solve[{g1, g2, g3}, {y, z, w}]

3

Remove["Global`*"]

x = 
      Log[Sqrt[(E^(3 ( Log[E^2] + Log[E^6])))/
            E^Log[3]]]

x // Simplify

N[%, 20]

x // N

4

Remove["Global`*"]

x = ((E^(1/2 Log[2] + Log[32]))^(1/
                3)) ((E^(1/3))^2)^Log[8]

x // Simplify

N[%, 20]

x // N

5

Remove["Global`*"]

A = 
      Log[a^(1/4)] - 1/2 Log[1/(b - Sqrt[b^2 - a^2])] + 
          1/2 Log[b/a + Sqrt[(b^2)/(a^2) - 1]] // 
        Simplify

b^2  ≥  a^2

Bedingungen:
1)   a >  0
2)  (b^2) > (a^2)
3) (b/a)> 0 , also  b > 0

Somit   0 < a < b

6

Remove["Global`*"]

γ = 32 Degree // N

a = 5.886;
b = 2.159;
c^2 == a^2 + b^2 - 2 a b Cos[γ] // Solve

c = 4.213371606707448`;
a/Sinα == c/Sin[γ] // Solve

Sinα = 0.7402871335419849`;  α = 
      ArcSin[Sinα]

α = Pi - α

α/Degree

b/Sinβ == c/Sin[γ] // Solve

Sinβ = 0.2715392322998888`;  β = 
      ArcSin[Sinβ]

β/Degree

Kontrolle

(α + β + γ)/Degree

7

Remove["Global`*"]

u = (1/a^3 - 1/b^3)/(1/a^2 + 1/(a b) + 1/b^2)

u // Simplify

res = ((u 4^(4 (x + 1)) 6^(-3 
                      x) 4^(-4) == 3^(2 - x))) // 
        Simplify

res // Solve

Lösung :   {{x -> -Log[
                    (1/9) ((1/a) - (1/b))]/(-2 Log[3] + 
                    Log[32])}}

Damit noch etwas herumprobieren:

N[%, 20]

-Log[1/9]/(-2 Log[3] + Log[32]) // N

-Log[(1/9) ((1/a) - (1/b))]/(-2 Log[3] + 
                Log[32]) + (-Log[9] + 
              Log[1/a - 1/b])/(-2 Log[3] + Log[32]) // 
      Simplify

8

Remove["Global`*"]

g1 = (u == (x + y)/Sqrt[2]);  
    g2 = (v == (x - y)/Sqrt[2]);
solv = Solve[{g1, g2}, {x, y}] // Flatten

(g3 = (y^2 - x^2 == 1) /. solv) // 
      FullSimplify

Solve[-2 u v == 1, {v}]

v[u_] := -(1/(2 u));  Plot[v[u], {u, -2, 2}];

Hyperbel

9

Remove["Global`*"]

res = (x - 
            3  y) (2  b + 
              2/3  a)/((1/9 a^2 + b^2 + 2/3  a b) (x^3 - 
                  27  y^3))

res // Simplify

10

Remove["Global`*"]

α = ArcSin[Tan[1]/10 Cos[Sin[Cos[1]]]] // N

β = ArcCos[Cot[1] Sin[Cos[Sin[1]]]] // N

alpha  kleiner als  beta.