91Ó°ÊÓ

Find an equation of variation in which: \(y\) varies jointly as \(w\) and the square of \(x\) and inversely as \(z,\) and \(y=49\) when \(w=3, x=7\) and \(z=12\).

Solve. $$ x^{2}=6+x $$

Write simplified for each of the following. Be sure to list all restrictions on the domain. $$ f(x)=\frac{x^{3}+4 x}{x^{2}-16} \div \frac{x^{2}+8 x+15}{x^{2}+x-20} $$

Find an equation of variation in which: \(y\) varies directly as \(x\) and inversely as \(w\) and the square of \(z,\) and \(y=4.5\) when \(x=15, w=5,\) and \(z=2\).

Solve. $$ \frac{1}{3}-x \leq \frac{2}{5} \quad $$

Astronomy. When two galaxies are moving in opposite directions at velocities \(v_{1}\) and \(v_{2},\) an observer in one of the galaxies would see the other galaxy receding at speed $$ \frac{v_{1}+v_{2}}{1+\frac{v_{1} v_{2}}{c^{2}}} $$ where \(c\) is the speed of light. Determine the observed speed if \(v_{1}\) and \(v_{2}\) are both one-fourth the speed of light.

Simplify. Do not use negative exponents in the answer. $$ \left(-4 x^{3}\right)^{0} $$

Perform the indicated operations and, if possible, simplify. $$ \frac{4 x^{2}-9 y^{2}}{8 x^{3}-27 y^{3}} \div \frac{4 x+6 y}{3 x-9 y} \cdot \frac{4 x^{2}+6 x y+9 y^{2}}{4 x^{2}-8 x y+3 y^{2}} $$

Find simplified form for the reciprocal of $$ x^{2}+x+1+\frac{1}{x}+\frac{1}{x^{2}} $$

Simplify. Do not use negative exponents in the answer. $$ \frac{24 a^{-4} c^{-8}}{16 a^{5} c^{-7}} $$