Answer:
See below for proof.
Step-by-step explanation:
Given radical expression:
[tex]\left(\sqrt[3]{3}\right)\left(\sqrt{3}\right)[/tex]
To express the given radical expression as ⁶√(243), we can use the laws of exponents.
[tex]\boxed{\begin{array}{rl}\underline{\sf Laws\;of\;Exponents}\\\\\sf Product:&a^m \times a^n=a^{m+n}\\\\\sf Power\;of\;a\;Power:&(a^m)^n=a^{mn}\\\\\sf Fractional\;Exponent:&a^{\frac{1}{n}}=\sqrt[n]{a}\\\\\end{array}}[/tex]
Apply the fractional exponent rule:
[tex]3^{\frac13}\cdot 3^{\frac12}[/tex]
Apply the product exponent rule, which states that when two terms with the same base are multiplied, we add their exponents:
[tex]3^{\frac13+\frac12}[/tex]
Simplify the exponent:
[tex]3^{\frac56}[/tex]
Rewrite the exponent as the product of 5 and ¹/₆:
[tex]3^{5\cdot {\frac16}}[/tex]
Apply the power of a power exponent rule:
[tex]\left(3^5\right)^{\frac16}[/tex]
Evaluate 3⁵:
[tex]\left(243\right)^{\frac16}[/tex]
Apply the fractional exponent rule:
[tex]\sqrt[6]{243}[/tex]
