Solving a usually involves three main steps.
Isolate the radical on one side of the equation.
Raise each side of the equation to a power equal to the index of the radical to eliminate the radical.
Solve the resulting equation.
Check the results for .
Now we can analyze the given radical equation.
8 10 x 3 − 15 = 17 \begin{gathered}
8\sqrt[3]{10x}-15=17
\end{gathered} 8 3 1 0 x − 1 5 = 1 7
First, let's isolate the radical,
10 x 3 , \sqrt[3]{10x}, 3 1 0 x , on one side of the equation.
8 10 x 3 − 15 = 17 8\sqrt[3]{10x}-15=17 8 3 1 0 x − 1 5 = 1 7 8 10 x 3 = 32 8\sqrt[3]{10x}=32 8 3 1 0 x = 3 2 10 x 3 = 4 \sqrt[3]{10x}=4 3 1 0 x = 4
We got an isolated radical with index equal to
3 . {\color{#0000FF}{3}}. 3 . Then, we will raise each side of the equation to the power of
3 . {\color{#0000FF}{3}}. 3 . 10 x 3 = 4 \sqrt[3]{10x}=4 3 1 0 x = 4 ( 10 x 3 ) 3 = 4 3 \left(\sqrt[3]{10x}\right)^3=4^3 ( 3 1 0 x ) 3 = 4 3 x = 64 10 x=\dfrac{64}{10} x = 1 0 6 4
x = 32 5 x=\dfrac{32}{5} x = 5 3 2
Next, we will check for extraneous solutions. We do that by substituting
32 5 \frac{32}{5} 5 3 2 for
x x x in the original equation. If the substitution produces a true statement, we know that our answer is correct. If it does not, then it is an extraneous solution.
8 10 x 3 − 15 = 17 8\sqrt[3]{10x}-15=17 8 3 1 0 x − 1 5 = 1 7 8 10 ( 32 5 ) 3 − 15 = ? 17 8\sqrt[3]{10\left({\color{#0000FF}{\dfrac{32}{5}}}\right)}-15\stackrel{?}{=}17 8 3 1 0 ( 5 3 2 ) − 1 5 = ? 1 7 8 320 5 3 − 15 = ? 17 8\sqrt[3]{\dfrac{320}{5}}-15\stackrel{?}{=}17 8 3 5 3 2 0 − 1 5 = ? 1 7 8 64 3 − 15 = ? 17 8\sqrt[3]{64}-15\stackrel{?}{=}17 8 3 6 4 − 1 5 = ? 1 7 8 ( 4 ) − 15 = ? 17 8(4)-15\stackrel{?}{=}17 8 ( 4 ) − 1 5 = ? 1 7 32 − 15 = ? 17 32-15\stackrel{?}{=}17 3 2 − 1 5 = ? 1 7
17 = 17 ✓ 17=17\ {\color{#009600}{\Large\checkmark}} 1 7 = 1 7 ✓
Because our substitution produced a true statement, we know that our answer,
x = 32 5 , x=\frac{32}{5}, x = 5 3 2 , is correct.
