Question

find the product with the exponent in simplest form. then, identify the values of \\(x\\) and \\(y\\).

\\(6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}} = 6^{\frac{x}{y}}\\)

\\(x = 7\\), \\(y = 12\\)

write the expression \\(6^{\frac{7}{12}}\\) in radical form.

\\(\bigcirc \sqrt7{6^{12}}\\)
\\(\bigcirc \sqrt12{6^7}\\)
\\(\bigcirc \sqrt12{7 \cdot 6}\\)

Explanation:

Simplify the product of exponential terms

Using the Exponent Rules knowledge point

$$ 6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}} = 6^{\frac{1}{3} + \frac{1}{4}} = 6^{\frac{4}{12} + \frac{3}{12}} = 6^{\frac{7}{12}} $$

Thus, comparing \(6^{\frac{7}{12}}\) to \(6^{\frac{x}{y}}\) yields \(x = 7\) and \(y = 12\).

Convert the exponential expression to radical form

Using the Radical to Exponential Form knowledge point

$$ 6^{\frac{7}{12}} = \sqrt[12]{6^7} $$

Answer:

Question 1

\(x = 7\), \(y = 12\)

Question 2

• (A) \(\sqrt[7]{6^{12}}\)
• (B) \(\sqrt[12]{6^7}\) (Correct answer)
• (C) \(\sqrt[12]{7 \cdot 6}\)