Question

$$(6^{7})^{3}=6^{21}$$

$$left(6^{\frac{1}{8}} ight)^{8}=6$$

let $x = 6^{\frac{1}{8}}$ $x^{8}=6$ $x=sqrt8{6}$ use the two different equations for $x$ above to express $6^{\frac{1}{8}}$ as a radical in the answer form below. $6^{\frac{1}{8}}=square$

Explanation:

Step1: Recall the definition of rational exponents

The formula for converting a rational exponent to a radical is \( a^{\frac{m}{n}}=\sqrt[n]{a^{m}} \) (when \( n>0 \)). For the expression \( 6^{\frac{1}{8}} \), we have \( m = 1 \) and \( n=8 \).

Step2: Apply the formula

Using the formula \( a^{\frac{1}{n}}=\sqrt[n]{a} \) (since \( m = 1 \)), when \( a = 6 \) and \( n=8 \), we get \( 6^{\frac{1}{8}}=\sqrt[8]{6} \). Also, from the given steps in the problem, we set \( x = 6^{\frac{1}{8}} \), found that \( x^{8}=6 \), and then solved for \( x \) to get \( x=\sqrt[8]{6} \), which confirms that \( 6^{\frac{1}{8}}=\sqrt[8]{6} \).

Answer:

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