Solve the given equation 3=(243^(1)/(3))(3^(x)/(3)) x=square (Type an integer or a simplified fraction.)

To solve the equation \(3 = (243^{\frac{1}{3}})(3^{\frac{x}{3}})\), we need to simplify both sides of the equation.<br /><br />First, let's simplify \(243^{\frac{1}{3}}\):<br />\[ 243^{\frac{1}{3}} = \sqrt[3]{243} \]<br /><br />Since \(243 = 3^5\), we have:<br />\[ \sqrt[3]{243} = \sqrt[3]{3^5} = 3^{5/3} \]<br /><br />Now, substitute this back into the original equation:<br />\[ 3 = (3^{5/3})(3^{\frac{x}{3}}) \]<br /><br />Next, use the property of exponents that states \(a^m \cdot a^n = a^{m+n}\):<br />\[ 3 = 3^{5/3 + \frac{x}{3}} \]<br /><br />Since the bases are the same, we can equate the exponents:<br />\[ 1 = \frac{5}{3} + \frac{x}{3} \]<br /><br />To solve for \(x\), first combine the fractions on the right-hand side:<br />\[ 1 = \frac{5 + x}{3} \]<br /><br />Multiply both sides by 3 to clear the fraction:<br />\[ 3 = 5 + x \]<br /><br />Finally, solve for \(x\):<br />\[ x = 3 - 5 \]<br />\[ x = -2 \]<br /><br />So, the solution to the equation is:<br />\[ x = -2 \]