Question

simplify the following expression.
$(-5)^{-5}$
$(-5)^{-5} = \square$ (type an integer or a fraction.)

Explanation:

Step1: Recall the negative exponent rule

The rule for negative exponents is \(a^{-n}=\frac{1}{a^{n}}\) (where \(a
eq0\) and \(n\) is a positive integer). So for \((-5)^{-5}\), we can apply this rule.
\((-5)^{-5}=\frac{1}{(-5)^{5}}\)

Step2: Calculate \((-5)^{5}\)

When raising a negative number to an odd power, the result is negative. \((-5)^{5}=(-5)\times(-5)\times(-5)\times(-5)\times(-5)\). Let's calculate step by step: \((-5)\times(-5) = 25\), \(25\times(-5)=-125\), \(-125\times(-5) = 625\), \(625\times(-5)=-3125\). So \((-5)^{5}=-3125\).

Step3: Substitute back into the fraction

We had \(\frac{1}{(-5)^{5}}\), and since \((-5)^{5}=-3125\), then \(\frac{1}{(-5)^{5}}=\frac{1}{-3125}=-\frac{1}{3125}\).

Answer:

\(-\frac{1}{3125}\)