Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x^2+5x+6=0
-
Equation 6x+1=-4x
- Equation (3x−36)⋅(x+8)=0
-
Equation 3*x^2+2*x-1=0
- Express {x} in terms of y where:
- 5*x-13*y=17
- 13*x-12*y=19
- -11*x-3*y=14
- -13*x+7*y=-15
- Sum of series:
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64
-
Identical expressions
- (one / two)^(fourteen -5x)= sixty-four
- (1 divide by 2) to the power of (14 minus 5x) equally 64
- (one divide by two) to the power of (fourteen minus 5x) equally sixty minus four
- (1/2)(14-5x)=64
- 1/214-5x=64
- 1/2^14-5x=64
- (1 divide by 2)^(14-5x)=64
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Similar expressions
- 16^sin^2(x)-16*4^(sqrt(3)*sin(2x))=0
- (1/2)^(14+5x)=64
(1/2)^(14-5x)=64 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$\left(\frac{1}{2}\right)^{14 - 5 x} = 64$$
or
$$\left(\frac{1}{2}\right)^{14 - 5 x} - 64 = 0$$
or
$$\frac{32^{x}}{16384} = 64$$
or
$$32^{x} = 1048576$$
- this is the simplest exponential equation
Do replacement
$$v = 32^{x}$$
we get
$$v - 1048576 = 0$$
or
$$v - 1048576 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = 1048576$$
We get the answer: v = 1048576
do backward replacement
$$32^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(32 \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(1048576 \right)}}{\log{\left(32 \right)}} = 4$$
$$\left(\frac{1}{2}\right)^{14 - 5 x} = 64$$
or
$$\left(\frac{1}{2}\right)^{14 - 5 x} - 64 = 0$$
or
$$\frac{32^{x}}{16384} = 64$$
or
$$32^{x} = 1048576$$
- this is the simplest exponential equation
Do replacement
$$v = 32^{x}$$
we get
$$v - 1048576 = 0$$
or
$$v - 1048576 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = 1048576$$
We get the answer: v = 1048576
do backward replacement
$$32^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(32 \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(1048576 \right)}}{\log{\left(32 \right)}} = 4$$
Rapid solution
[src]
x1 = 4
$$x_{1} = 4$$
log(1048576) 4*pi*I
x2 = ------------ - --------
5*log(2) 5*log(2)
$$x_{2} = \frac{\log{\left(1048576 \right)}}{5 \log{\left(2 \right)}} - \frac{4 i \pi}{5 \log{\left(2 \right)}}$$
log(1048576) 2*pi*I
x3 = ------------ - --------
5*log(2) 5*log(2)
$$x_{3} = \frac{\log{\left(1048576 \right)}}{5 \log{\left(2 \right)}} - \frac{2 i \pi}{5 \log{\left(2 \right)}}$$
log(1048576) 2*pi*I
x4 = ------------ + --------
5*log(2) 5*log(2)
$$x_{4} = \frac{\log{\left(1048576 \right)}}{5 \log{\left(2 \right)}} + \frac{2 i \pi}{5 \log{\left(2 \right)}}$$
log(1048576) 4*pi*I
x5 = ------------ + --------
5*log(2) 5*log(2)
$$x_{5} = \frac{\log{\left(1048576 \right)}}{5 \log{\left(2 \right)}} + \frac{4 i \pi}{5 \log{\left(2 \right)}}$$
x5 = log(1048576)/(5*log(2)) + 4*i*pi/(5*log(2))
Sum and product of roots
[src]
sum
log(1048576) 4*pi*I log(1048576) 2*pi*I log(1048576) 2*pi*I log(1048576) 4*pi*I
4 + ------------ - -------- + ------------ - -------- + ------------ + -------- + ------------ + --------
5*log(2) 5*log(2) 5*log(2) 5*log(2) 5*log(2) 5*log(2) 5*log(2) 5*log(2)
$$\left(\left(\left(4 + \left(\frac{\log{\left(1048576 \right)}}{5 \log{\left(2 \right)}} - \frac{4 i \pi}{5 \log{\left(2 \right)}}\right)\right) + \left(\frac{\log{\left(1048576 \right)}}{5 \log{\left(2 \right)}} - \frac{2 i \pi}{5 \log{\left(2 \right)}}\right)\right) + \left(\frac{\log{\left(1048576 \right)}}{5 \log{\left(2 \right)}} + \frac{2 i \pi}{5 \log{\left(2 \right)}}\right)\right) + \left(\frac{\log{\left(1048576 \right)}}{5 \log{\left(2 \right)}} + \frac{4 i \pi}{5 \log{\left(2 \right)}}\right)$$
=
4*log(1048576)
4 + --------------
5*log(2)
$$4 + \frac{4 \log{\left(1048576 \right)}}{5 \log{\left(2 \right)}}$$
product
/log(1048576) 4*pi*I \ /log(1048576) 2*pi*I \ /log(1048576) 2*pi*I \ /log(1048576) 4*pi*I \ 4*|------------ - --------|*|------------ - --------|*|------------ + --------|*|------------ + --------| \ 5*log(2) 5*log(2)/ \ 5*log(2) 5*log(2)/ \ 5*log(2) 5*log(2)/ \ 5*log(2) 5*log(2)/
$$4 \left(\frac{\log{\left(1048576 \right)}}{5 \log{\left(2 \right)}} - \frac{4 i \pi}{5 \log{\left(2 \right)}}\right) \left(\frac{\log{\left(1048576 \right)}}{5 \log{\left(2 \right)}} - \frac{2 i \pi}{5 \log{\left(2 \right)}}\right) \left(\frac{\log{\left(1048576 \right)}}{5 \log{\left(2 \right)}} + \frac{2 i \pi}{5 \log{\left(2 \right)}}\right) \left(\frac{\log{\left(1048576 \right)}}{5 \log{\left(2 \right)}} + \frac{4 i \pi}{5 \log{\left(2 \right)}}\right)$$
=
2 4
256*pi 256*pi
1024 + --------- + -----------
2 4
5*log (2) 625*log (2)
$$\frac{256 \pi^{4}}{625 \log{\left(2 \right)}^{4}} + 1024 + \frac{256 \pi^{2}}{5 \log{\left(2 \right)}^{2}}$$
1024 + 256*pi^2/(5*log(2)^2) + 256*pi^4/(625*log(2)^4)
Numerical answer
[src]
x1 = 4.0
x2 = 4.0 - 3.62588811346175*i
x3 = 4.0 - 1.81294405673088*i
x4 = 4.0 + 1.81294405673088*i
x5 = 4.0 + 3.62588811346175*i
x5 = 4.0 + 3.62588811346175*i