Let's start with the given equation:
[tex]\[ y = \sqrt[3]{27x - 54} + 5 \][/tex]
We are asked to rewrite this equation in the form:
[tex]\[ y = a \sqrt[3]{x - h} + k \][/tex]
In order to reach this form, we need to manipulate the expression inside the cube root:
1. First, let's factor out 27 from the expression inside the cube root:
[tex]\[ \sqrt[3]{27x - 54} \][/tex]
2. Recognize that [tex]\(54 = 27 \times 2\)[/tex], so the expression can be rewritten as:
[tex]\[ \sqrt[3]{27(x - 2)} \][/tex]
3. Since [tex]\(27 = 3^3\)[/tex], we use the property of cube roots [tex]\(\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}\)[/tex] to simplify:
[tex]\[ \sqrt[3]{27(x - 2)} = \sqrt[3]{3^3 \cdot (x - 2)} = 3 \sqrt[3]{x - 2} \][/tex]
Thus, the equation is now:
[tex]\[ y = 3 \sqrt[3]{x - 2} + 5 \][/tex]
Our goal was to rewrite the original equation in the form [tex]\( y = a \sqrt[3]{x - h} + k \)[/tex]. Comparing:
[tex]\[ y = 3 \sqrt[3]{x - 2} + 5 \][/tex]
with
[tex]\[ y = a \sqrt[3]{x - h} + k \][/tex]
we identify that:
- [tex]\(a = 3\)[/tex]
- [tex]\(h = 2\)[/tex]
- [tex]\(k = 5\)[/tex]
Now, let’s check which of the given options match this form:
1. [tex]\( y = -27 \sqrt[3]{x + 2} + 5 \)[/tex]
2. [tex]\( y = -3 \sqrt[3]{x + 2} + 5 \)[/tex]
3. [tex]\( y = 3 \sqrt[3]{x - 2} + 5 \)[/tex]
4. [tex]\( y = 27 \sqrt[3]{x - 2} + 5 \)[/tex]
The correct equivalent equation is:
[tex]\[ y = 3 \sqrt[3]{x - 2} + 5 \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{3} \][/tex]
