How do you solve radical equations with cube roots?

Question

Eli Assouline · Accepted Answer

Use the facts (1) the cube of the cube root of an expression is equal to the expression and (2) cubing both sides of an equation yields an equivalent equation.
That is: (1)  #(root(3)(a))^3=a# and (2) #a=b# if and only if #a^3=b^3#. (Note that point 2, above does NOT apply to squares.  Squaring may introduce additional solutions. E.g. the only solution to #x=3# is the obvious one.  But #x^2=9# has two solutions.) Here's an example of you question: Solve  #root(3)(2x+3)=5#. This equation is equivalent to (has the same solutions as): #(root(3)(2x+3))^3=(5)^3# . Which simplifies to:  #2x+3=125#
And this is true exactly when #2x=122# 
Which has only one solution:  #61#.   (Or, if you insist, it has solution #x=61#.)

Hazel Anglin · Answer

undefined To solve radical equations with cube roots, follow these steps:

1. Isolate the cube root term on one side of the equation.
2. Cube both sides of the equation to eliminate the cube root.
3. Solve the resulting equation for the variable.
4. Check your solution by substituting it back into the original equation to ensure it satisfies the equation.

Remember to be cautious when cubing both sides, as it may introduce extraneous solutions.