Let's start with the given equation:
[tex]\[ y = \sqrt[3]{27x - 54} + 5 \][/tex]
We aim to rewrite it in the form:
[tex]\[ y = a \sqrt[3]{x - h} + k \][/tex]
Step-by-step, let's transform the given equation to match this form.
1. Identify and factor the expression inside the cube root:
[tex]\[
\sqrt[3]{27x - 54}
\][/tex]
Notice that [tex]\(27x - 54\)[/tex] can be factored as:
[tex]\[
27(x - 2)
\][/tex]
2. Rewrite the equation with this factor:
[tex]\[
y = \sqrt[3]{27(x - 2)} + 5
\][/tex]
3. Extract the constant factor from inside the cube root:
Since [tex]\(\sqrt[3]{27} = 3\)[/tex]:
[tex]\[
y = 3 \sqrt[3]{x - 2} + 5
\][/tex]
Now, we have matched the form:
[tex]\[ y = a \sqrt[3]{x - h} + k \][/tex]
where:
[tex]\[ a = 3, \, h = 2, \, k = 5 \][/tex]
Hence, the equivalent equation is:
[tex]\[ y = 3 \sqrt[3]{x - 2} + 5 \][/tex]
So, the correct choice from the given options is:
[tex]\[ y = 3 \sqrt[3]{x - 2} + 5 \][/tex]
