Evaluate \( y=\left(\frac{1}{5}\right)^{x} \), when \( x=4 \) \( y=625 \) \( y=-625 \) \( y=\frac{1}{625} \)

To evaluate \( y=\left(\frac{1}{5}\right)^{x} \) when \( x=4 \), we substitute \( x \) with 4: \[ y=\left(\frac{1}{5}\right)^{4} = \frac{1^{4}}{5^{4}} = \frac{1}{625} \] So, the correct answer is \( y=\frac{1}{625} \). Now, just for fun, here’s a deeper dive into the magical world of exponents: Whenever you raise a fraction to an exponent, it's like raising both the numerator and the denominator to that exponent. This means \( \left(\frac{1}{5}\right)^{4} \) transforms into \( \frac{1^{4}}{5^{4}} \). And since \( 5^{4} = 625 \), you end up with \( \frac{1}{625} \). Would you believe that in mathematics, exponents have fascinating rules? One common mistake is forgetting that a negative exponent means you should take the reciprocal. So, for example, \( a^{-n} = \frac{1}{a^{n}} \). Always remember: negative exponents can be turned into fractions, making it easier to handle those tricky calculations!