To solve the equation [tex]\(7 = \sqrt[3]{2 - x} = 12\)[/tex], let's follow a logical step-by-step approach:
1. Isolate the cube root term:
We can rewrite the equation focusing on isolating the cube root:
[tex]\[
7 = \sqrt[3]{2 - x}
\][/tex]
This implies:
[tex]\[
\sqrt[3]{2 - x} = 12 - 7
\][/tex]
Simplifying the right side of the equation:
[tex]\[
\sqrt[3]{2 - x} = 5
\][/tex]
2. Eliminate the cube root by cubing both sides of the equation:
To remove the cube root, we cube both sides:
[tex]\[
\left(\sqrt[3]{2 - x}\right)^3 = 5^3
\][/tex]
This simplifies to:
[tex]\[
2 - x = 125
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
We need to isolate [tex]\(x\)[/tex] by rearranging the equation:
[tex]\[
2 - 125 = x
\][/tex]
Simplifying the left side:
[tex]\[
-123 = x
\][/tex]
Therefore, the solution to the equation is:
[tex]\[
x = -123
\][/tex]
