In exercises 1 through 7, find the value of the expression:
-
\[
(81)^{1/4}
\]
-
\[
8^{4/3}
\]
-
\[
(25)^{3/2}
\]
-
\[
(25)^{-3/2}
\]
-
\[
\left( \frac{1}{8} \right)^{-2/3}
\]
-
\[
\left( \frac{27}{16} \right)^{-1/2}
\]
-
\[
(0.01)^{-1}
\]
In exercises 8 through 13, simplify the expression:
-
\[
\left( \cfrac{ \mathrm{e}^7 }{ \mathrm{e}^3 } \right)^{-1}
\]
-
\[
\cfrac{ 3^3 3^5 }{ \left( 3^4 \right)^3 }
\]
-
\[
\cfrac{ 5^{1/2} \left( 5^{1/2} \right)^5 }{5^4}
\]
-
\[
\cfrac{ 2^{-3}2^5 }{ \left( 2^4 \right)^{-3} }
\]
-
\[
\cfrac{ \left( 2^4 \right)^{1/3} }{ 16 \left( 2^{7/3} \right) }
\]
-
\[
\cfrac{ \left( 2^{1/3} 3^{2/3} \right)^3 }{ 3^{5/2} 3^{-1/2} }
\]
In exercises 14 through 19, solve the equation.
-
\[
2^{2x-1} = 8
\]
-
\[
\left( \frac{1}{3} \right)^{x+1} = 27
\]
-
\[
8 \sqrt[3]{2} = 4^x
\]
-
\[
\left( 3^{2x}\, 3^2 \right)^4 = 3
\]
-
\[
\mathrm{e}^{-6x+1} = \mathrm{e}^3
\]
-
\[
\mathrm{e}^{x^{2}-2x} = \mathrm{e}^3
\]
In exercises 20 through 28, sketch the graph of the functions using translation and reflection techniques in all of them, except for exercises 25 and 27.
-
\[
y = \mathrm{e}^{x+2}
\]
-
\[
y = -2\mathrm{e}^x +1
\]
-
\[
y = \mathrm{e}^{-x}
\]
-
\[
y = \mathrm{e}^{-x}+2
\]
-
\[
y = 2-\mathrm{e}^{-x}
\]
-
\[
y = 3^x
\]
-
\[
y = 3^{-x+2}
\]
-
\[
y = 4^x
\]
-
\[
y = -4^{-x-1}
\]
-
If \(g(x)=A\mathrm{e}^{-kx}\), \(g(0)=9\) and \(g(2)=5\), find \(g(6)\).
-
If \(h(x)=30-P\mathrm{e}^{-kx}\), \(h(0)=10\) and \(h(3)=-30\), find \(h(12)\).
