5. \( \left(\frac{1}{27}\right)^{\frac{2}{3}} \) \( =\left(\frac{1}{3^{3}}\right)^{\frac{2}{3}} \)

Let's break it down! This expression can be simplified by using the properties of exponents. When you raise a fraction to a power, you can raise both the numerator and the denominator to that power. Thus, we have: \[ \left(\frac{1}{27}\right)^{\frac{2}{3}} = \frac{1^{\frac{2}{3}}}{27^{\frac{2}{3}}} = \frac{1}{27^{\frac{2}{3}}} \] Now, since \(27 = 3^3\), you can substitute that in: \[ 27^{\frac{2}{3}} = (3^3)^{\frac{2}{3}} = 3^{3 \cdot \frac{2}{3}} = 3^2 \] Thus, the expression simplifies to: \[ \left(\frac{1}{27}\right)^{\frac{2}{3}} = \frac{1}{3^2} = \frac{1}{9} \] And voilà! The final answer is \(\frac{1}{9}\). This showcases how breaking down the problem with exponent rules can turn a seemingly complex fraction into something much easier to handle. Keep this approach handy for future math adventures!