-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

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.

r+13=0

Subtract 13 from both sides of the equation.

r=-13

r=-13

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