Question

use the equation \\(y = \sqrt3{27x - 54} + 5\\).
which is an equivalent equation of the form \\(y = a\sqrt3{x - h} + k\\)?

\\(y = -27\sqrt3{x + 2} + 5\\)
\\(y = -3\sqrt3{x + 2} + 5\\)
\\(y = 3\sqrt3{x - 2} + 5\\)
\\(y = 27\sqrt3{x - 2} + 5\\)

Explanation:

Factor the radicand

Using the Radical Simplification knowledge point

$$ 27x - 54 = 27(x - 2) $$

Apply the product rule for radicals

Using the Radical Simplification knowledge point

$$ \sqrt[3]{27(x - 2)} = \sqrt[3]{27} \cdot \sqrt[3]{x - 2} $$

Simplify the cube root of the constant

Using the Radical Simplification knowledge point

$$ \sqrt[3]{27} = 3 $$

Substitute back into the original equation

Using the Cube Root Transformations knowledge point

$$ y = 3\sqrt[3]{x - 2} + 5 $$

Answer:

• \(y = -27\sqrt[3]{x + 2} + 5\)
• \(y = -3\sqrt[3]{x + 2} + 5\)
• \(y = 3\sqrt[3]{x - 2} + 5\) (Correct answer)
• \(y = 27\sqrt[3]{x - 2} + 5\)