# How do you solve radical equations with cube roots?

Mar 6, 2015

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) ${\left(\sqrt[3]{a}\right)}^{3} = a$ 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 $\sqrt[3]{2 x + 3} = 5$.

This equation is equivalent to (has the same solutions as): ${\left(\sqrt[3]{2 x + 3}\right)}^{3} = {\left(5\right)}^{3}$ .

Which simplifies to: $2 x + 3 = 125$
And this is true exactly when $2 x = 122$
Which has only one solution: $61$.

(Or, if you insist, it has solution $x = 61$.)