Chapter 11

Simplify. Leave your answers as improper fractions. $$\frac{4 a^{2}-4 x^{2}}{\frac{a+x}{a-x}}$$

Challenge Problems.$$a^{8}-b^{8}$$

Factor completely, by hand or by calculator. Check your results. Trinomials with a Leading Coefficient of 1. $$b^{2}+b-12$$

Reduce to lowest terms. Write your answers without negative exponents. Do some algebraic fractions by calculator. $$\frac{36}{44}$$

Sum or Difference of Two Cubes. $$1-64 y^{3}$$

$$a^{2} c+b^{2} c+c^{2} d$$

Multiply and reduce. Do some by calculator. $$\frac{x+y}{10} \cdot \frac{a x}{3(x+y)}$$

Solve for \(x .\) Try some by calculator. $$a x-a b=c x-b c$$

Solve for \(x\). Assume the integers in these equations to be exact numbers, and leave your answers in fractional form. \(3 x-\frac{x}{6}+\frac{x}{12}=70\)

Combine and simplify. Try some by calculator. $$\frac{1}{a}+\frac{5}{a}$$

Simplify. Leave your answers as improper fractions. $$\frac{x^{2}-\frac{y^{2}}{2}}{\frac{x-3 y}{2}}$$

Challenge Problems.$$4 m^{2}-9 n^{4}$$

Factor completely, by hand or by calculator. Check your results. Trinomials with a Leading Coefficient of 1. $$b^{2}-b-12$$

Reduce to lowest terms. Write your answers without negative exponents. Do some algebraic fractions by calculator. $$\frac{2 a b}{6 b}$$

$$4 x^{2} y+c x y^{2}+3 x y^{3}$$

Multiply and reduce. Do some by calculator. $$\frac{c}{x^{2}-y^{2}} \cdot \frac{d}{x^{2}-y^{2}}$$

Solve for \(x\). Assume the integers in these equations to be exact numbers, and leave your answers in fractional form. \(\frac{x}{4}+\frac{x}{6}+\frac{x}{8}=26\)

Combine and simplify. Try some by calculator. $$\frac{3}{x}+\frac{2}{x}-\frac{1}{x}$$

Simplify. Leave your answers as improper fractions. $$\frac{\frac{a b}{7}-3 d}{3 c-\frac{a b}{d}}$$

Challenge Problems.$$9^{2}-4 b^{4}$$

Reduce to lowest terms. Write your answers without negative exponents. Do some algebraic fractions by calculator. $$\frac{12 m^{2} n}{15 m n^{2}}$$

Sum or Difference of Two Cubes. $$a^{3}-27$$

$$4 a b x+6 a^{2} x^{2}+8 a x$$

Divide and reduce. Try some by calculator. $$\frac{7}{9} \div \frac{5}{3}$$

Solve for \(x\). Assume the integers in these equations to be exact numbers, and leave your answers in fractional form. \(\frac{6 x-19}{2}=\frac{2 x-11}{3}\)

Combine and simplify. Try some by calculator. $$\frac{2 a}{y}+\frac{3}{y}-\frac{a}{y}$$

Simplify. Leave your answers as improper fractions. $$\frac{1+\frac{1}{x+1}}{1-\frac{1}{x-1}}$$

Reduce to lowest terms. Write your answers without negative exponents. Do some algebraic fractions by calculator. $$\frac{21 m^{2} p^{2}}{28 m p^{4}}$$

Sum or Difference of Two Cubes. $$x^{3}-1$$

Divide and reduce. Try some by calculator. $$3 \frac{7}{8} \div 2$$

Solve for \(x\). Assume the integers in these equations to be exact numbers, and leave your answers in fractional form. \(\frac{7 x-40}{8}=\frac{9 x-80}{10}\)

Combine and simplify. Try some by calculator. $$\frac{x}{3 a}-\frac{y}{3 a}+\frac{z}{3 a}$$

Simplify. Leave your answers as improper fractions. $$\frac{x y-\frac{3 x}{a c}}{\frac{a c}{x}+2 c}$$

Challenge Problems.$$9 a^{2} b^{2}-4 c^{4}$$

$$2 a^{2} c-2 a^{2} c^{2}+3 a c$$

Reduce to lowest terms. Write your answers without negative exponents. Do some algebraic fractions by calculator. $$\frac{a b x-b x^{2}}{a c x-c x^{2}}$$

Sum or Difference of Two Cubes. $$x^{3}-64$$

Solve for \(x\). Assume the integers in these equations to be exact numbers, and leave your answers in fractional form. \(\frac{3 x-116}{4}+\frac{180-5 x}{6}=0\)

Combine and simplify. Try some by calculator. $$\frac{5 x}{2}-\frac{3 x}{2}$$

A car travels a distance \(d_{1}\) at a rate \(V_{1},\) then another distance \(d_{2}\) at a rate \(V_{2}\) The average speed for the entire trip is $$\text { average speed }=\frac{d_{1}+d_{2}}{\frac{d_{1}}{V_{1}}+\frac{d_{2}}{V_{2}}}$$ Simplify this complex fraction.

Challenge Problems.$$25 x^{4}-16 y^{6}$$

$$5 a c d-2 c^{2} d^{2}+b c d$$

Factor completely, by hand or by calculator. Check your results. The General Quadratic Trinomial. $$4 x^{2}-13 x+3$$

Reduce to lowest terms. Write your answers without negative exponents. Do some algebraic fractions by calculator. $$\frac{x^{2}-4}{x^{3}-8}$$

Sum or Difference of Two Cubes. $$x^{3}+1$$

Divide and reduce. Try some by calculator. $$\frac{9}{16} \div 8$$

Solve for \(x\). Assume the integers in these equations to be exact numbers, and leave your answers in fractional form. \(\frac{3 x-4}{2}-\frac{3 x-1}{16}=\frac{6 x-5}{8}\)

Combine and simplify. Try some by calculator. $$\frac{7}{x+2}-\frac{5}{x+2}$$

The equivalent resistance of two resistors in parallel is $$\frac{R_{1} R_{2}}{R_{1}+R_{2}}$$ If each resistor is made of wire of resistivity \(\rho,\) with \(R_{1}\) using a wire of length \(L_{1}\) and cross-sectional area \(A_{1},\) and \(R_{2}\) having a length \(L_{2}\) and area \(A_{2}\) our expression becomes $$\frac{\frac{\rho L_{1}}{A_{1}} \cdot \frac{\rho L_{2}}{A_{2}}}{\frac{\rho L_{1}}{A_{1}}+\frac{\rho L_{2}}{A_{2}}}$$ Simplify this complex fraction.

Challenge Problems.$$36 y^{2}-49 z^{6}$$