Question

solve.
\sqrt3{42x} = -6

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the solution(s) is(are) x =
(type an integer or a simplified fraction. use a comma to separate answers as needed.)

b. there is no solution.

Explanation:

Step1: Cube both sides to eliminate the cube root

To solve the equation \(\sqrt[3]{42x} = -6\), we cube both sides of the equation. Cubing a cube root cancels out the cube root operation. So, we have:

$$ (\sqrt[3]{42x})^3 = (-6)^3 $$

Step2: Simplify both sides

Simplifying the left side, \((\sqrt[3]{42x})^3\) simplifies to \(42x\) because cubing a cube root of a number gives the number itself. On the right side, \((-6)^3 = -6\times -6\times -6 = -216\). So now our equation is:

$$ 42x = -216 $$

Step3: Solve for x

To solve for \(x\), we divide both sides of the equation by 42. So,

$$ x = \frac{-216}{42} $$

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 6. Dividing -216 by 6 gives -36, and dividing 42 by 6 gives 7. So,

$$ x = -\frac{36}{7} $$

Answer:

The solution is \(x = -\frac{36}{7}\), so the correct choice is A. The solution(s) is(are) \(x = \boldsymbol{-\frac{36}{7}}\).