Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation sin(x)=cos(x)
-
Equation cos(x/2)=1/2
-
Equation sin(x)=√3/2
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Equation sin(x)=1,5
- Express {x} in terms of y where:
- xy^1=y+x*sin(y/x)
- sqrt(x)*sin(y/x)=0
- 12*10^(-30)-0,1884*10^12*x^2-y(0,3768*10^12*x+1,183152*10^12*x^3)+y^2(2,760688*10^24*x^2)=0=0
- (x^2+y^2)×(x+y-3)=2xy
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Identical expressions
- nine ^(-3y)=- nine ^(-5x)
- 9 to the power of ( minus 3y) equally minus 9 to the power of ( minus 5x)
- nine to the power of ( minus 3y) equally minus nine to the power of ( minus 5x)
- 9(-3y)=-9(-5x)
- 9-3y=-9-5x
- 9^-3y=-9^-5x
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Similar expressions
- 9^(-3y)=-9^(5x)
- 9^(-3y)=+9^(-5x)
- 9^(3y)=-9^(-5x)
9^(-3y)=-9^(-5x) equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$9^{- 3 y} = - 9^{- 5 x}$$
or
$$9^{- 3 y} + 9^{- 5 x} = 0$$
or
$$\left(\frac{1}{59049}\right)^{x} = - 9^{- 3 y}$$
or
$$\left(\frac{1}{59049}\right)^{x} = - 9^{- 3 y}$$
- this is the simplest exponential equation
Do replacement
$$v = \left(\frac{1}{59049}\right)^{x}$$
we get
$$v + 9^{- 3 y} = 0$$
or
$$v + 9^{- 3 y} = 0$$
do backward replacement
$$\left(\frac{1}{59049}\right)^{x} = v$$
or
$$x = - \frac{\log{\left(v \right)}}{\log{\left(59049 \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(- 9^{- 3 y} \right)}}{\log{\left(\frac{1}{59049} \right)}} = - \frac{\log{\left(- 729^{- y} \right)}}{\log{\left(59049 \right)}}$$
$$9^{- 3 y} = - 9^{- 5 x}$$
or
$$9^{- 3 y} + 9^{- 5 x} = 0$$
or
$$\left(\frac{1}{59049}\right)^{x} = - 9^{- 3 y}$$
or
$$\left(\frac{1}{59049}\right)^{x} = - 9^{- 3 y}$$
- this is the simplest exponential equation
Do replacement
$$v = \left(\frac{1}{59049}\right)^{x}$$
we get
$$v + 9^{- 3 y} = 0$$
or
$$v + 9^{- 3 y} = 0$$
do backward replacement
$$\left(\frac{1}{59049}\right)^{x} = v$$
or
$$x = - \frac{\log{\left(v \right)}}{\log{\left(59049 \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(- 9^{- 3 y} \right)}}{\log{\left(\frac{1}{59049} \right)}} = - \frac{\log{\left(- 729^{- y} \right)}}{\log{\left(59049 \right)}}$$
The graph
Rapid solution
[src]
/ 6*re(y)\ / 6*y\
log\3 / I*arg\-3 /
x1 = ------------- + ------------
10*log(3) 10*log(3)
$$x_{1} = \frac{\log{\left(3^{6 \operatorname{re}{\left(y\right)}} \right)}}{10 \log{\left(3 \right)}} + \frac{i \arg{\left(- 3^{6 y} \right)}}{10 \log{\left(3 \right)}}$$
/| _______|\ / _______\
||10/ 6*y || | 10/ 6*y |
log\|\/ -3 |/ I*arg\-\/ -3 /
x2 = ----------------- + ------------------
log(3) log(3)
$$x_{2} = \frac{\log{\left(\left|{\sqrt[10]{- 3^{6 y}}}\right| \right)}}{\log{\left(3 \right)}} + \frac{i \arg{\left(- \sqrt[10]{- 3^{6 y}} \right)}}{\log{\left(3 \right)}}$$
/| _______ _______ _______ ____________|\
|| 10/ 6*y ___ 10/ 6*y ___ 10/ 6*y / ___ ||
|| \/ -3 \/ 5 *\/ -3 \/ 2 *\/ -3 *\/ -5 - \/ 5 || / _______ / ___________\\
log||- ---------- + ---------------- - --------------------------------|| |10/ 6*y | ___ ___ / ___ ||
\| 4 4 4 |/ I*arg\\/ -3 *\-1 + \/ 5 - I*\/ 2 *\/ 5 + \/ 5 //
x3 = ------------------------------------------------------------------------- + -------------------------------------------------------
log(3) log(3)
$$x_{3} = \frac{\log{\left(\left|{- \frac{\sqrt[10]{- 3^{6 y}}}{4} + \frac{\sqrt{5} \sqrt[10]{- 3^{6 y}}}{4} - \frac{\sqrt{2} \sqrt[10]{- 3^{6 y}} \sqrt{-5 - \sqrt{5}}}{4}}\right| \right)}}{\log{\left(3 \right)}} + \frac{i \arg{\left(\sqrt[10]{- 3^{6 y}} \left(-1 + \sqrt{5} - \sqrt{2} i \sqrt{\sqrt{5} + 5}\right) \right)}}{\log{\left(3 \right)}}$$
/| _______ _______ _______ ____________|\
|| 10/ 6*y ___ 10/ 6*y ___ 10/ 6*y / ___ ||
|| \/ -3 \/ 5 *\/ -3 \/ 2 *\/ -3 *\/ -5 - \/ 5 || / _______ / ___________\\
log||- ---------- + ---------------- + --------------------------------|| |10/ 6*y | ___ ___ / ___ ||
\| 4 4 4 |/ I*arg\\/ -3 *\-1 + \/ 5 + I*\/ 2 *\/ 5 + \/ 5 //
x4 = ------------------------------------------------------------------------- + -------------------------------------------------------
log(3) log(3)
$$x_{4} = \frac{\log{\left(\left|{- \frac{\sqrt[10]{- 3^{6 y}}}{4} + \frac{\sqrt{5} \sqrt[10]{- 3^{6 y}}}{4} + \frac{\sqrt{2} \sqrt[10]{- 3^{6 y}} \sqrt{-5 - \sqrt{5}}}{4}}\right| \right)}}{\log{\left(3 \right)}} + \frac{i \arg{\left(\sqrt[10]{- 3^{6 y}} \left(-1 + \sqrt{5} + \sqrt{2} i \sqrt{\sqrt{5} + 5}\right) \right)}}{\log{\left(3 \right)}}$$
/| _______ _______ _______ ____________|\
||10/ 6*y ___ 10/ 6*y ___ 10/ 6*y / ___ ||
||\/ -3 \/ 5 *\/ -3 \/ 2 *\/ -3 *\/ -5 + \/ 5 || / _______ / ____________\\
log||---------- + ---------------- - --------------------------------|| |10/ 6*y | ___ ___ / ___ ||
\| 4 4 4 |/ I*arg\\/ -3 *\1 + \/ 5 - \/ 2 *\/ -5 + \/ 5 //
x5 = ----------------------------------------------------------------------- + -----------------------------------------------------
log(3) log(3)
$$x_{5} = \frac{\log{\left(\left|{\frac{\sqrt[10]{- 3^{6 y}}}{4} + \frac{\sqrt{5} \sqrt[10]{- 3^{6 y}}}{4} - \frac{\sqrt{2} \sqrt[10]{- 3^{6 y}} \sqrt{-5 + \sqrt{5}}}{4}}\right| \right)}}{\log{\left(3 \right)}} + \frac{i \arg{\left(\sqrt[10]{- 3^{6 y}} \left(1 + \sqrt{5} - \sqrt{2} \sqrt{-5 + \sqrt{5}}\right) \right)}}{\log{\left(3 \right)}}$$
/| _______ _______ _______ ____________|\
||10/ 6*y ___ 10/ 6*y ___ 10/ 6*y / ___ ||
||\/ -3 \/ 5 *\/ -3 \/ 2 *\/ -3 *\/ -5 + \/ 5 || / _______ / ____________\\
log||---------- + ---------------- + --------------------------------|| |10/ 6*y | ___ ___ / ___ ||
\| 4 4 4 |/ I*arg\\/ -3 *\1 + \/ 5 + \/ 2 *\/ -5 + \/ 5 //
x6 = ----------------------------------------------------------------------- + -----------------------------------------------------
log(3) log(3)
$$x_{6} = \frac{\log{\left(\left|{\frac{\sqrt[10]{- 3^{6 y}}}{4} + \frac{\sqrt{5} \sqrt[10]{- 3^{6 y}}}{4} + \frac{\sqrt{2} \sqrt[10]{- 3^{6 y}} \sqrt{-5 + \sqrt{5}}}{4}}\right| \right)}}{\log{\left(3 \right)}} + \frac{i \arg{\left(\sqrt[10]{- 3^{6 y}} \left(1 + \sqrt{5} + \sqrt{2} \sqrt{-5 + \sqrt{5}}\right) \right)}}{\log{\left(3 \right)}}$$
/| _______ _______ _______ ___________|\
||10/ 6*y ___ 10/ 6*y ___ 10/ 6*y / ___ ||
||\/ -3 \/ 5 *\/ -3 I*\/ 2 *\/ -3 *\/ 5 - \/ 5 || / _______ / ___________\\
log||---------- + ---------------- + ---------------------------------|| | 10/ 6*y | ___ ___ / ___ ||
\| 4 4 4 |/ I*arg\-\/ -3 *\1 + \/ 5 + I*\/ 2 *\/ 5 - \/ 5 //
x7 = ------------------------------------------------------------------------ + -------------------------------------------------------
log(3) log(3)
$$x_{7} = \frac{\log{\left(\left|{\frac{\sqrt[10]{- 3^{6 y}}}{4} + \frac{\sqrt{5} \sqrt[10]{- 3^{6 y}}}{4} + \frac{\sqrt{2} i \sqrt[10]{- 3^{6 y}} \sqrt{5 - \sqrt{5}}}{4}}\right| \right)}}{\log{\left(3 \right)}} + \frac{i \arg{\left(- \sqrt[10]{- 3^{6 y}} \left(1 + \sqrt{5} + \sqrt{2} i \sqrt{5 - \sqrt{5}}\right) \right)}}{\log{\left(3 \right)}}$$
/| _______ _______ _______ ____________|\
|| 10/ 6*y ___ 10/ 6*y ___ 10/ 6*y / ___ ||
|| \/ -3 \/ 5 *\/ -3 \/ 2 *\/ -3 *\/ -5 + \/ 5 || / _______ / ____________\\
log||- ---------- - ---------------- + --------------------------------|| |10/ 6*y | ___ ___ / ___ ||
\| 4 4 4 |/ I*arg\\/ -3 *\-1 - \/ 5 + \/ 2 *\/ -5 + \/ 5 //
x8 = ------------------------------------------------------------------------- + ------------------------------------------------------
log(3) log(3)
$$x_{8} = \frac{\log{\left(\left|{- \frac{\sqrt{5} \sqrt[10]{- 3^{6 y}}}{4} - \frac{\sqrt[10]{- 3^{6 y}}}{4} + \frac{\sqrt{2} \sqrt[10]{- 3^{6 y}} \sqrt{-5 + \sqrt{5}}}{4}}\right| \right)}}{\log{\left(3 \right)}} + \frac{i \arg{\left(\sqrt[10]{- 3^{6 y}} \left(- \sqrt{5} - 1 + \sqrt{2} \sqrt{-5 + \sqrt{5}}\right) \right)}}{\log{\left(3 \right)}}$$
/| _______ _______ _______ ____________|\
||10/ 6*y ___ 10/ 6*y ___ 10/ 6*y / ___ ||
||\/ -3 \/ 5 *\/ -3 \/ 2 *\/ -3 *\/ -5 - \/ 5 || / _______ / ___________\\
log||---------- - ---------------- - --------------------------------|| |10/ 6*y | ___ ___ / ___ ||
\| 4 4 4 |/ I*arg\\/ -3 *\1 - \/ 5 - I*\/ 2 *\/ 5 + \/ 5 //
x9 = ----------------------------------------------------------------------- + ------------------------------------------------------
log(3) log(3)
$$x_{9} = \frac{\log{\left(\left|{- \frac{\sqrt{5} \sqrt[10]{- 3^{6 y}}}{4} + \frac{\sqrt[10]{- 3^{6 y}}}{4} - \frac{\sqrt{2} \sqrt[10]{- 3^{6 y}} \sqrt{-5 - \sqrt{5}}}{4}}\right| \right)}}{\log{\left(3 \right)}} + \frac{i \arg{\left(\sqrt[10]{- 3^{6 y}} \left(- \sqrt{5} + 1 - \sqrt{2} i \sqrt{\sqrt{5} + 5}\right) \right)}}{\log{\left(3 \right)}}$$
/| _______ _______ _______ ____________|\
||10/ 6*y ___ 10/ 6*y ___ 10/ 6*y / ___ ||
||\/ -3 \/ 5 *\/ -3 \/ 2 *\/ -3 *\/ -5 - \/ 5 || / _______ / ___________\\
log||---------- - ---------------- + --------------------------------|| |10/ 6*y | ___ ___ / ___ ||
\| 4 4 4 |/ I*arg\\/ -3 *\1 - \/ 5 + I*\/ 2 *\/ 5 + \/ 5 //
x10 = ----------------------------------------------------------------------- + ------------------------------------------------------
log(3) log(3)
$$x_{10} = \frac{\log{\left(\left|{- \frac{\sqrt{5} \sqrt[10]{- 3^{6 y}}}{4} + \frac{\sqrt[10]{- 3^{6 y}}}{4} + \frac{\sqrt{2} \sqrt[10]{- 3^{6 y}} \sqrt{-5 - \sqrt{5}}}{4}}\right| \right)}}{\log{\left(3 \right)}} + \frac{i \arg{\left(\sqrt[10]{- 3^{6 y}} \left(- \sqrt{5} + 1 + \sqrt{2} i \sqrt{\sqrt{5} + 5}\right) \right)}}{\log{\left(3 \right)}}$$
x10 = log(Abs(-sqrt(5)*(-3^(6*y))^(1/10)/4 + (-3^(6*y))^(1/10)/4 + sqrt(2)*(-3^(6*y))^(1/10)*sqrt(-5 - sqrt(5))/4))/log(3) + i*arg((-3^(6*y))^(1/10)*(-sqrt(5) + 1 + sqrt(2)*i*sqrt(sqrt(5) + 5)))/log(3)