SOLUTION: I would like to know if I did the first three correct and I need help on #4: Solve these equations: (a) 2x^-3=16 1/x^3=8 1=8x^3 1/8=x^3 x=1/2 (b)2x^-3=0 x^-3=0 1/x^3=0

Question 556154: I would like to know if I did the first three correct and I need help on #4:
Solve these equations:
(a) 2x^-3=16
1/x^3=8
1=8x^3
1/8=x^3
x=1/2
(b)2x^-3=0
x^-3=0
1/x^3=0
x=0
(c) 2^-3x=16
-3x=4
x=-4/3
(d) cuberoot 5=25^2x
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
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(a) is good.
to confirm, use your calculator to get:
2*(1/2)^-3 = 16
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(b) doesn't look good.
i don't think you have a solution for that problem.
you start with:
2x^(-3) = 0
this is equivalent to:
2/(x^(3) = 0
multiply both sides of this equation by x^3 to get:
2 = 0 which is not true.
if you said that x = 0, you would not get a solution.
example:
2*x^(-3) = 0 becomes:
2*0^(-3) = 0 becomes:
2/0^3) = 0 becomes:
undefined because you can't divide by 0.
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(c) 2^(-3x) = 16
this becomes:
1/(2^3x) = 16
divide both sides of this equation by 16 and multiply both sides of this equation by 2^3x to get:
2^3x = 1/16
take the log of both sides of this equation to get:
log(2^3x) = log(1/16)
this becomes:
3x*log(2) = log(1/16)
divide both sides of this equation by 3*log(2) to get:
x = log(1/16)/(3*log(2)) which becomes:
x = -1.33333
confirm by replacing x in your original equation:
2^(-3x) = 16 becomes:
2^(-3*-1.33333333) = 16 which becomes:
2^(4) = 16 which becomes:
16 = 16
this confirms the value of x = -1.33333333 is good.
note that 1.333333333 is really 1+1/3 = 4/3.
2^(-3x) = 16 becomes:
2^(-3*(-4/3) which becomes:
2^4 = 16 which becomes:
16 = 16
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(d)
cube root (5)=25^2x becomes:
5^(1/3) = 25^2x
take the log of both sides to get:
log(5^(1/3)() = log(25^2x)
this becomes:
(1/3)*log(5) = 2*x*log(25)
divide both sides of this equation by 2*log(25) to get:
x = (1/3)*log(5) / (2*log(25) which becomes:
x = log(5) / (3*2*log(25))
solve for x to get:
x = .08333333333
confirm by substituting in original equation.
5^(1/3) = 25^(2x) becomes:
1.709975947 = 25^(2*.0833333333) which becomes:
1.709975947 = 1.709975947
this confirms the value for x is good.

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