To find an equivalent equation of the form [tex]\( y = a \sqrt[3]{x - h} + k \)[/tex] from the given equation [tex]\( y = \sqrt[3]{27x - 54} + 5 \)[/tex], we need to follow a series of steps to simplify and transform the given equation.
1. Identify the given equation:
[tex]\[ y = \sqrt[3]{27x - 54} + 5 \][/tex]
2. Rewrite the expression inside the cube root by factoring:
Notice that [tex]\( 27x - 54 \)[/tex] can be factored as [tex]\( 27(x - 2) \)[/tex]:
[tex]\[ 27x - 54 = 27(x - 2) \][/tex]
3. Substitute this back into the given equation:
[tex]\[ y = \sqrt[3]{27(x - 2)} + 5 \][/tex]
4. Utilize the property of cube roots:
Recall that [tex]\( \sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b} \)[/tex]. Apply this to [tex]\( \sqrt[3]{27(x - 2)} \)[/tex]:
[tex]\[ \sqrt[3]{27(x - 2)} = \sqrt[3]{27} \cdot \sqrt[3]{x - 2} \][/tex]
5. Calculate the cube root of 27:
[tex]\[ \sqrt[3]{27} = 3 \][/tex]
6. Put it all together:
Substitute [tex]\( \sqrt[3]{27} \)[/tex] back into the equation:
[tex]\[ \sqrt[3]{27(x - 2)} = 3 \cdot \sqrt[3]{x - 2} \][/tex]
7. Form the final equivalent equation:
[tex]\[ y = 3 \cdot \sqrt[3]{x - 2} + 5 \][/tex]
Thus, the equivalent equation of the form [tex]\( y = a \sqrt[3]{x - h} + k \)[/tex] is:
[tex]\[ y = 3 \sqrt[3]{x - 2} + 5 \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{y = 3 \sqrt[3]{x - 2} + 5} \][/tex]
