Question

Use the properties of exponents to simplify the expressions. (a) \(\left(\frac{1}{e}\right)^{-2}\) (b) \(\left(\frac{e^{5}}{e^{2}}\right)^{-1}\) (c) \(e^{0}\) (d) \(\frac{1}{e^{-3}}\)

Short answer

(a) \(e^{2}\) \n (b) \(\frac{1}{e^{3}}\) \n (c) 1 \n (d) \(e^{3}\)

Step-by-step solution

- Simplifying (a)

Here, we have \(\left(\frac{1}{e}\right)^{-2}\). By the properties of negative exponents, this can be rewritten as \((e)^{2}\), which simply equals \(e^{2}\).

- Simplifying (b)

For this one, we have \(\left(\frac{e^{5}}{e^{2}}\right)^{-1}\). Applying the property of exponents for division first, this becomes \(\left(e^{5-2}\right)^{-1} = (e^{3})^{-1}\). Further simplifying using the properties of negative exponents, we finally have \(\frac{1}{e^{3}}\).

- Simplifying (c)

Simplifying \(e^{0}\) is straightforward. Any non-zero number raised to the power of 0 is 1. Therefore, \(e^{0} = 1\).

- Simplifying (d)

Lastly, we simplify \(\frac{1}{e^{-3}}\). For this case, we use the negative exponent property which leads us to rewrite the expression as \(e^{3}\).