Sure! Let's simplify each expression step-by-step using the properties of exponents.
(a) [tex]\(\left(\frac{1}{e}\right)^{-2}\)[/tex]
To simplify this expression, we can use the property of exponents [tex]\(\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n\)[/tex].
[tex]\[
\left(\frac{1}{e}\right)^{-2} = \left(\frac{e}{1}\right)^2 = e^2
\][/tex]
Thus, the simplified form of [tex]\(\left(\frac{1}{e}\right)^{-2}\)[/tex] is [tex]\(e^2\)[/tex].
[tex]\[
\boxed{7.38905609893065}
\][/tex]
(b) [tex]\(\left(\frac{e^5}{e^3}\right)^{-1}\)[/tex]
First, simplify the fraction inside the parentheses using the property [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex].
[tex]\[
\frac{e^5}{e^3} = e^{5-3} = e^2
\][/tex]
Next, apply the exponent -1 to this result using the property [tex]\(\left(a^m\right)^{-1} = \frac{1}{a^m}\)[/tex].
[tex]\[
\left(e^2\right)^{-1} = \frac{1}{e^2}
\][/tex]
Thus, the simplified form of [tex]\(\left(\frac{e^5}{e^3}\right)^{-1}\)[/tex] is [tex]\(\frac{1}{e^2}\)[/tex].
[tex]\[
\boxed{0.1353352832366127}
\][/tex]
(c) [tex]\(\left(e^5\right)\left(e^2\right)\)[/tex]
To simplify this expression, use the property [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex].
[tex]\[
e^5 \cdot e^2 = e^{5+2} = e^7
\][/tex]
Thus, the simplified form of [tex]\(e^5 \cdot e^2\)[/tex] is [tex]\(e^7\)[/tex].
[tex]\[
\boxed{1096.6331584284585}
\][/tex]
(d) [tex]\(\frac{e^0}{e^{-5}}\)[/tex]
First, recall that any number raised to the power of 0 is 1, [tex]\(e^0 = 1\)[/tex]. So we can rewrite the expression as:
[tex]\[
\frac{1}{e^{-5}}
\][/tex]
Next, use the property [tex]\(\frac{1}{a^{-n}} = a^n\)[/tex].
[tex]\[
\frac{1}{e^{-5}} = e^5
\][/tex]
Thus, the simplified form of [tex]\(\frac{e^0}{e^{-5}}\)[/tex] is [tex]\(e^5\)[/tex].
[tex]\[
\boxed{148.4131591025766}
\][/tex]
These are the simplified forms of the given expressions.
