Question

what is the solution to the equation (sqrt3{m - 4} - sqrt3{2m + 8} = 0)?

(m = -24)

(m = -12)

(m = -6)

(m = -4)

Explanation:

Step1: Isolate one cube root

Start with the equation \(\sqrt[3]{m - 4}-\sqrt[3]{2m + 8}=0\). Add \(\sqrt[3]{2m + 8}\) to both sides:
\(\sqrt[3]{m - 4}=\sqrt[3]{2m + 8}\)

Step2: Eliminate cube roots

Since the cube root function is one - to - one (if \(\sqrt[3]{a}=\sqrt[3]{b}\), then \(a = b\)), we can cube both sides of the equation \(\sqrt[3]{m - 4}=\sqrt[3]{2m + 8}\). Cubing the left side gives \((\sqrt[3]{m - 4})^3=m - 4\), and cubing the right side gives \((\sqrt[3]{2m + 8})^3=2m + 8\). So we have the equation:
\(m-4 = 2m + 8\)

Step3: Solve for \(m\)

Subtract \(m\) from both sides:
\(m - m-4=2m - m + 8\)
\(-4=m + 8\)
Then subtract 8 from both sides:
\(-4-8=m+8 - 8\)
\(m=-12\)
We can check our solution by plugging \(m = - 12\) back into the original equation:
Left side: \(\sqrt[3]{-12 - 4}-\sqrt[3]{2\times(-12)+8}=\sqrt[3]{-16}-\sqrt[3]{-24 + 8}=\sqrt[3]{-16}-\sqrt[3]{-16}=0\)
Right side: \(0\)
So the solution is valid.

Answer:

\(m=-12\) (corresponding to the option "m = -12")