Question

this module is intended to help you understand fractional exponents. rewrite the expression below as 6 to a single power:
$(6^4)^9 = 6^{36}$
$(6^7)^3 = 6^{21}$
$(6^{\frac{1}{8}})^8 = 6$
let $x = 6^{\frac{1}{8}}$
$x^8 = 6$
solve for a positive value of $x$, expressing your answer with a radical symbol.
$x = \square$

Explanation:

Step1: Recall the definition of radicals and exponents

We know that if \(x^{n}=a\) (where \(n\) is a positive integer and we want the positive root), then \(x = \sqrt[n]{a}\). Also, from the equation \(x^{8}=6\), we need to solve for \(x\) (positive value).

Step2: Apply the nth - root definition

Given \(x^{8}=6\), to find \(x\) (positive), we take the 8th root of both sides. By the definition of the \(n\)th root, if \(x^{n}=a\), then \(x=\sqrt[n]{a}\) for positive \(x\) when \(n\) is a positive integer. Here \(n = 8\) and \(a = 6\), so \(x=\sqrt[8]{6}\). Also, we know that \(x = 6^{\frac{1}{8}}\) from the given \(x = 6^{\frac{1}{8}}\), and if we raise both sides to the power of 1 (or use the inverse operation of raising to the 8th power), taking the 8th root of 6 is equivalent to \(6^{\frac{1}{8}}\), but in radical form, it is \(\sqrt[8]{6}\).

Answer:

\(\sqrt[8]{6}\)