evaluate each expression.\na. $729^{\\frac{5}{6}}$\n$729^{\\frac{5}{6}} = \\left(\\sqrt6{729}\\right)^5$\n$=…

evaluate each expression.\na. $729^{\\frac{5}{6}}$\n$729^{\\frac{5}{6}} = \\left(\\sqrt6{729}\\right)^5$\n$= \\left(\\sqrt6{\\underline{\\quad\\quad\\quad\\quad\\quad\\quad}}\\right)^5$\n$= \\left(\\sqrt6{\\underline{\\quad}}\\right)^5$\n$= \\underline{\\quad\\quad}$ or $\\underline{\\quad\\quad}$\n$b^{\\frac{m}{n}} = \\left(\\sqrtn{b}\\right)^m$\n$729 = 3\\cdot3\\cdot3\\cdot3\\cdot3\\cdot3$\n$3\\cdot3\\cdot3\\cdot3\\cdot3\\cdot3 = 3^6$\n$\\sqrt6{3^6} = 3$

evaluate each expression.\na. $729^{\\frac{5}{6}}$\n$729^{\\frac{5}{6}} = \\left(\\sqrt6{729}\\right)^5$\n$= \\left(\\sqrt6{\\underline{\\quad\\quad\\quad\\quad\\quad\\quad}}\\right)^5$\n$= \\left(\\sqrt6{\\underline{\\quad}}\\right)^5$\n$= \\underline{\\quad\\quad}$ or $\\underline{\\quad\\quad}$\n$b^{\\frac{m}{n}} = \\left(\\sqrtn{b}\\right)^m$\n$729 = 3\\cdot3\\cdot3\\cdot3\\cdot3\\cdot3$\n$3\\cdot3\\cdot3\\cdot3\\cdot3\\cdot3 = 3^6$\n$\\sqrt6{3^6} = 3$

Answer

Explanation:

Step1: Substitute 729 with prime factorization

$729^{\frac{5}{6}} = \left(\sqrt[6]{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}\right)^5$

Step2: Rewrite as exponential form

$= \left(\sqrt[6]{3^6}\right)^5$

Step3: Simplify the 6th root

$= \left(3\right)^5$

Step4: Calculate the final power

$3^5 = 243$ Alternatively, rewrite the original expression as $\left(729^{\frac{1}{6}}\right)^5 = 3^5 = 243$, or use $729^{\frac{5}{6}} = (3^6)^{\frac{5}{6}} = 3^{6 \times \frac{5}{6}} = 3^5 = 243$

Answer:

$243$