Algebra Word Problem Solutions

Simplify each of the following expressions. Be sure that your answer has no negative or fractional exponents. $a\cdot (\frac{1}{81}{)}^{-\frac{1}{4}}b\cdot x^{-2}{y}^{-4}c\cdot (2x{)}^{-2}(16x^{2y}{)}^{\frac{1}{2}}$

Rewrite the following as an equivalent logarithmic equation. ${\displaystyle e^3=20.0855}$ An equivalent logarithmic for $e^3=20.0855$ is ?

Consider a challenge in capital budgeting where six projects are represented by $0-1\text{variables}{x}_1,x_2,x_3,x_4,x_5,\text{and}{x}_6.$
a. Write a constraint modeling a situation in which two of the projects 1, 3, and 6 must be undertaken.
b. In which situation the constraint "$x_3-x_5=0$" is used, explain clearly:
c. Create a constraint to represent a scenario where either roject 2 or 4 must be pursued, but not both
d. Write constraints modeling a situation where project 1 cannot be undertaken IF projects 3. also is NOT undertaken.
e. Explain clearly the situation in which the following 3 constraints are used simulataneously (together):
${\displaystyle x_4\le x_1}$
${\displaystyle x_4\le x_3}$
${\displaystyle x_4\ge x_1+x_3-1}$

Let $x_1=94,210,x_2=8631,x_3=1440,x_4=133,$ and $x5=34.$ Calculate each of the following, using four-digit decimal floating-point arithmetic: $x1+((x2+x3)+(x4+x5))$