# Solve for r -1/27=r^3

-127=r3
Rewrite the equation as r3=-127.
r3=-127
Move 127 to the left side of the equation by adding it to both sides.
r3+127=0
Factor the left side of the equation.
Rewrite 1 as 13.
r3+1327=0
Rewrite 27 as 33.
r3+1333=0
Rewrite 1333 as (13)3.
r3+(13)3=0
Since both terms are perfect cubes, factor using the sum of cubes formula, a3+b3=(a+b)(a2-ab+b2) where a=r and b=13.
(r+13)(r2-r13+(13)2)=0
Simplify.
Combine 13 and r.
(r+13)(r2-r3+(13)2)=0
Apply the product rule to 13.
(r+13)(r2-r3+1232)=0
One to any power is one.
(r+13)(r2-r3+132)=0
Raise 3 to the power of 2.
(r+13)(r2-r3+19)=0
(r+13)(r2-r3+19)=0
(r+13)(r2-r3+19)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
r+13=0
r2-r3+19=0
Set the first factor equal to 0 and solve.
Set the first factor equal to 0.
r+13=0
Subtract 13 from both sides of the equation.
r=-13
r=-13
Set the next factor equal to 0 and solve.
Set the next factor equal to 0.
r2-r3+19=0
Multiply through by the least common denominator 9, then simplify.
Apply the distributive property.
9r2+9(-r3)+9(19)=0
Simplify.
Cancel the common factor of 3.
Move the leading negative in -r3 into the numerator.
9r2+9(-r3)+9(19)=0
Factor 3 out of 9.
9r2+3(3)(-r3)+9(19)=0
Cancel the common factor.
9r2+3⋅(3(-r3))+9(19)=0
Rewrite the expression.
9r2+3(-r)+9(19)=0
9r2+3(-r)+9(19)=0
Multiply -1 by 3.
9r2-3r+9(19)=0
Cancel the common factor of 9.
Cancel the common factor.
9r2-3r+9(19)=0
Rewrite the expression.
9r2-3r+1=0
9r2-3r+1=0
9r2-3r+1=0
9r2-3r+1=0
Use the quadratic formula to find the solutions.
-b±b2-4(ac)2a
Substitute the values a=9, b=-3, and c=1 into the quadratic formula and solve for r.
3±(-3)2-4⋅(9⋅1)2⋅9
Simplify.
Simplify the numerator.
Raise -3 to the power of 2.
r=3±9-4⋅(9⋅1)2⋅9
Multiply 9 by 1.
r=3±9-4⋅92⋅9
Multiply -4 by 9.
r=3±9-362⋅9
Subtract 36 from 9.
r=3±-272⋅9
Rewrite -27 as -1(27).
r=3±-1⋅272⋅9
Rewrite -1(27) as -1⋅27.
r=3±-1⋅272⋅9
Rewrite -1 as i.
r=3±i⋅272⋅9
Rewrite 27 as 32⋅3.
Factor 9 out of 27.
r=3±i⋅9(3)2⋅9
Rewrite 9 as 32.
r=3±i⋅32⋅32⋅9
r=3±i⋅32⋅32⋅9
Pull terms out from under the radical.
r=3±i⋅(33)2⋅9
Move 3 to the left of i.
r=3±3i32⋅9
r=3±3i32⋅9
Multiply 2 by 9.
r=3±3i318
Simplify 3±3i318.
r=1±i36
r=1±i36
The final answer is the combination of both solutions.
r=1+i36,1-i36
r=1+i36,1-i36
The final solution is all the values that make (r+13)(r2-r3+19)=0 true.
r=-13,1+i36,1-i36
Solve for r -1/27=r^3

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