To transform the given equation [tex]\( y = \sqrt[3]{27x - 54} + 5 \)[/tex] into an equivalent equation of the form [tex]\( y = a \sqrt[3]{x - h} + k \)[/tex], we can follow these steps:
1. Identify and rewrite the term inside the cube root:
Given [tex]\( y = \sqrt[3]{27x - 54} + 5 \)[/tex], we recognize that the term inside the cube root, [tex]\( 27x - 54 \)[/tex], can be rewritten in a factored form.
2. Factor the expression inside the cube root:
[tex]\[
27x - 54 = 27(x - 2)
\][/tex]
3. Substitute the factored expression back into the original equation:
[tex]\[
y = \sqrt[3]{27(x - 2)} + 5
\][/tex]
4. Simplify the cube root:
Since [tex]\( 27 \)[/tex] is a perfect cube ([tex]\( 27 = 3^3 \)[/tex]), we can simplify the cube root:
[tex]\[
\sqrt[3]{27(x - 2)} = \sqrt[3]{27} \cdot \sqrt[3]{x - 2} = 3 \sqrt[3]{x - 2}
\][/tex]
5. Rewrite the equation with the simplified cube root:
[tex]\[
y = 3 \sqrt[3]{x - 2} + 5
\][/tex]
Therefore, the equivalent equation in the form [tex]\( y = a \sqrt[3]{x - h} + k \)[/tex] is:
[tex]\[
y = 3 \sqrt[3]{x - 2} + 5
\][/tex]
Among the given options, the correct choice is:
[tex]\[
y = 3 \sqrt[3]{x - 2} + 5
\][/tex]
