Algebra Review

Algebra Review: Learning Skills Center

Exponents & Radicals

• Exponents: Shorthand for repeated multiplication. In b^n, b is the base, and n is the exponent.

Properties of Exponents:

1. Whole Number Exponents: b^n = b \cdot b \cdot … \cdot b (n times)

   • Example: 2^3 = 2 \cdot 2 \cdot 2 = 8

2. Zero Exponent: b^0 = 1, where b \neq 0

   • Example: 5^0 = 1

3. Negative Exponents: b^{-n} = \frac{1}{b^n}, where b \neq 0

   • Example: 2^{-2} = \frac{1}{2^2} = \frac{1}{4}

4. Rational Exponents (nth root): \sqrt[n]{b} = b^{\frac{1}{n}}, where if n is even, then b \geq 0

   • Example: \sqrt[3]{8} = 8^{\frac{1}{3}} = 2

5. Rational Exponents: \sqrt[n]{b^m} = (\sqrt[n]{b})^m = (b^{\frac{1}{n}})^m = b^{\frac{m}{n}}, where if n is even, then b \geq 0

   • Example: \sqrt[3]{8^2} = (\sqrt[3]{8})^2 = 2^2 = 4

Operations with Exponents:

1. Multiplying Like Bases: b^n \cdot b^m = b^{n+m}. Add exponents.

   • Example: 2^2 \cdot 2^3 = 2^{2+3} = 2^5 = 32

2. Dividing Like Bases: \frac{b^n}{b^m} = b^{n-m}. Subtract exponents.

   • Example: \frac{3^5}{3^2} = 3^{5-2} = 3^3 = 27

3. Exponent of Exponent: (b^n)^m = b^{n \cdot m}. Multiply exponents.

   • Example: (2^3)^2 = 2^{3 \cdot 2} = 2^6 = 64

4. Removing Parentheses: (ab)^n = a^n \cdot b^n

   • Example: (2x)^3 = 2^3 \cdot x^3 = 8x^3

5. Removing Parentheses in Fractions: (\frac{a}{b})^n = \frac{a^n}{b^n}

   • Example: (\frac{x}{3})^2 = \frac{x^2}{3^2} = \frac{x^2}{9}

Special Conventions:

• -b^n = -(b^n)

• -b^n \neq (-b)^n

• k b^n = k(b^n)

• k b^n \neq (kb)^n

• (b)^n m = b^{(n m)}

• b^{n^m} \neq (b^n)^m

Logarithms

• Logarithms: Inverse of exponentiation.

  • If 2^3 = 8, then \log_2 8 = 3

• Definition: \log_b A = n if and only if b^n = A.

• Base b must be positive, so A is always positive.

• Common bases: 10 and e \approx 2.718

  • Base 10: \log A = n \Leftrightarrow 10^n = A

  • Example: \log 100 = 2 \Leftrightarrow 10^2 = 100

  • Natural logarithm: \ln A = n \Leftrightarrow e^n = A

  • Example: \ln e = 1 \Leftrightarrow e^1 = e

Good to Know:

• \log_b 1 = 0

• \log_b b = 1

Inverse Properties of Logs:

• \log_b b^x = x

• b^{\log_b x} = x

Laws of Logarithms:

1. Product Rule: \logb x + \logb y = \log_b (x \cdot y)

   • Example: \log2 4 + \log2 8 = \log2 (4 \cdot 8) = \log2 32 = 5

2. Quotient Rule: \logb x - \logb y = \log_b (\frac{x}{y})

   • Example: \log3 27 - \log3 3 = \log3 (\frac{27}{3}) = \log3 9 = 2

3. Power Rule: n \cdot \logb x = \logb x^n

   • Example: 2 \cdot \log5 5 = \log5 5^2 = \log_5 25 = 2

Factoring

• Factoring: Expressing a polynomial as a product.

• a \cdot b = 0 has solutions a = 0 or b = 0.

Special Products and Factoring Techniques

1. Distributive Law: ax + ay = a(x + y)

   • Example: 3x + 6y = 3(x + 2y)

2. Simple Trinomial: x^2 + (a+b)x + a \cdot b = (x + a)(x + b)

   • Example: x^2 + 5x + 6 = (x + 2)(x + 3)

3. Difference of Squares: x^2 – a^2 = (x – a)(x + a)

   • Example: x^2 – 16 = (x – 4)(x + 4)

4. Difference of Fourth Powers: x^4 – a^4 = (x^2 – a^2)(x^2 + a^2) = (x – a)(x + a)(x^2 + a^2)

   • Example: x^4 – 81 = (x^2 – 9)(x^2 + 9) = (x – 3)(x + 3)(x^2 + 9)

5. Sum or Difference of Cubes:

   • Sum: x^3 + a^3 = (x + a)(x^2 – ax + a^2)

     • Example: x^3 + 8 = (x + 2)(x^2 – 2x + 4)

   • Difference: x^3 – a^3 = (x – a)(x^2 + ax + a^2)

     • Example: x^3 – 27 = (x – 3)(x^2 + 3x + 9)

6. Factoring by Grouping: acx^3 + adx^2 + bcx + bd = ax^2(cx + d) + b(cx + d) = (ax^2 + b)(cx + d)

   • Example: 2x^3 – 3x^2 + 4x – 6 = x^2(2x – 3) + 2(2x – 3) = (x^2 + 2)(2x – 3)

7. Quadratic Formula: For ax^2 + bx + c = 0, x = \frac{–b \pm \sqrt{b^2 – 4ac}}{2a}

   • Example: x^2 + 2x – 1 = 0 \Rightarrow x = \frac{–2 \pm \sqrt{2^2 – 4(1)(–1)}}{2(1)} = \frac{–2 \pm \sqrt{8}}{2} = -1 \pm \sqrt{2}

Algebraic Fractions

• Rational expressions: Algebraic fractions with polynomials in numerator and denominator.

Operations with Fractions

1. Add Fractions: \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

   • Example: \frac{1}{2} + \frac{2}{3} = \frac{1 \cdot 3 + 2 \cdot 2}{2 \cdot 3} = \frac{3 + 4}{6} = \frac{7}{6}

2. Subtract Fractions: \frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}

   • Example: \frac{3}{4} - \frac{1}{3} = \frac{3 \cdot 3 - 1 \cdot 4}{4 \cdot 3} = \frac{9 - 4}{12} = \frac{5}{12}

3. Multiply Fractions: (\frac{a}{b})(\frac{c}{d}) = \frac{ac}{bd}

   • Example: (\frac{2}{3})(\frac{3}{5}) = \frac{2 \cdot 3}{3 \cdot 5} = \frac{6}{15} = \frac{2}{5}

4. Divide Fractions: Invert and multiply or multiply by the common denominator.

   • Invert and Multiply: \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc}

     • Example: \frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \cdot \frac{4}{3} = \frac{4}{6} = \frac{2}{3}

   • Multiply by Common Denominator: \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{\frac{a}{b}}{\frac{c}{d}} \cdot \frac{bd}{bd} = \frac{ad}{bc}

     • Example: \frac{\frac{1}{2}}{\frac{3}{4}} = \frac{\frac{1}{2}}{\frac{3}{4}} \cdot \frac{8}{8} = \frac{4}{6} = \frac{2}{3}

5. Cancel: \frac{ab}{ad} = \frac{b}{d}

   • Example: \frac{2x}{2y} = \frac{x}{y}

Rationalizing

a. If numerator or denominator is \sqrt{a}, multiply by \frac{\sqrt{a}}{\sqrt{a}}.
b. If numerator or denominator is \sqrt{a} - b, multiply by \frac{\sqrt{a}+b}{\sqrt{a}+b}.
c. If numerator or denominator is \sqrt{a} + b, multiply by \frac{\sqrt{a}-b}{\sqrt{a}-b}.

Equations

• Solving equations: Finding all solutions or roots.

• Solutions are unchanged by adding, subtracting, multiplying, or dividing both sides by the same quantity (except dividing by zero).

• Solution method depends on the degree of the equation.

• First-degree equations: Solve using addition, subtraction, multiplication, and division.

  • Example: 2x + 5 = 9 \Rightarrow 2x = 4 \Rightarrow x = 2

• Second-degree equations (quadratic equations): Solve by factoring or the quadratic formula.

  • Example: x^2 – 5x + 6 = 0 \Rightarrow (x – 2)(x – 3) = 0 \Rightarrow x = 2 \text{ or } x = 3

  • Quadratic formula: For ax^2 + bx + c = 0, x = \frac{–b \pm \sqrt{b^2 – 4ac}}{2a}

    • Example: x^2 – 5x + 6 = 0 \Rightarrow x = \frac{5 \pm \sqrt{(–5)^2 – 4(1)(6)}}{2(1)} = \frac{5 \pm \sqrt{1}}{2} = \frac{5 \pm 1}{2} \Rightarrow x = 3 \text{ or } x = 2

• Equations involving absolute value: Equivalent to two equations without the absolute value sign.

  • Example: |x + 2| = 5 \Rightarrow x + 2 = 5 \text{ or } –(x + 2) = 5 \Rightarrow x = 3 \text{ or } x = –7

Inequalities

• Inequalities: Mathematical expressions of comparison.

  • x > 3 (x is greater than 3)

  • -2 \leq a < 4 (a is between -2 (inclusive) and 4 (exclusive))

• Symbols:

  • >: greater than

  • <: less than

  • \geq: greater than or equal to

  • \leq: less than or equal to

• Algebraic inequalities: Generally have an infinite number of real solutions.

• Algebraic equations: Have a finite number of real solutions.

Solving Linear Inequalities

• Similar to linear equations, but multiplying or dividing by a negative number reverses the inequality sign.

  • Example: -2x + 3 < 7 \Rightarrow -2x < 4 \Rightarrow x > -2

Solving Absolute Value Inequalities

• Consider two cases: positive and negative values inside the absolute value.

  • If the value is negative, multiply by -1 to make it positive.

  • Example: |x - 3| \geq 4

    • Case I: (x - 3) is positive:

      • x - 3 \geq 4 \Rightarrow x \geq 7

    • Case II: (x - 3) is negative, make it positive:

      • -(x - 3) \geq 4 \Rightarrow x - 3 \leq -4 \Rightarrow x \leq -1

Solving Higher Order Inequalities

• Use a number line to account for all possible combinations of positive and negative terms.

  • Example: x^2 – 2x – 8 \geq 0 \Rightarrow (x – 4)(x + 2) \geq 0

    • Roots: x = -2, 4

    • Place roots on a number line, using closed circles for \geq or \leq.

    • Test each region to determine the sign of each term; the inequality sign determines the solution region.

---Begin Practice Problems---

Exponents and Radicals Practice

Simplify the following as much as possible:

1. 3^2 \cdot 3^4 \cdot 3^{-5} = 3^{2+4-5} = 3^1 = 3

2. (\frac{-2x^2}{y^3})^3 = \frac{(-2)^3 (x^2)^3}{(y^3)^3} = \frac{-8x^6}{y^9} = -\frac{8x^6}{y^9}

3. (\frac{7}{7^2})^{-1} = (7^{-1})^{-1} = 7

4. \frac{25 \cdot 5^2}{3} = \frac{25 \cdot 25}{3} = \frac{625}{3}

5. x(x^2)^n = x \cdot x^{2n} = x^{2n+1}

6. \frac{3x^2y^5}{9x^7y} = \frac{3}{9} \cdot \frac{x^2}{x^7} \cdot \frac{y^5}{y} = \frac{1}{3}x^{-5}y^4 = \frac{y^4}{3x^5}

7. (\frac{2x}{3y^4})^{\frac{2}{xy^7}} = \frac{4x^2}{9y^8}

8. 125^{\frac{1}{3}} = \sqrt[3]{125} = 5

9. (\sqrt{\frac{81}{100}})^{-\frac{3}{2}} = (\frac{9}{10})^{-\frac{3}{2}} = (\frac{10}{9})^{\frac{3}{2}} = \frac{10 \sqrt{10}}{9 \sqrt{9}} = \frac{10 \sqrt{10}}{27}

10. \sqrt[3]{\frac{-27r^6}{125s^9}} = \frac{\sqrt[3]{-27r^6}}{\sqrt[3]{125s^9}} = \frac{-3r^2}{5s^3} = -\frac{3r^2}{5s^3}

11. (\sqrt[3]{-8})^2 = (-2)^2 = 4

12. \sqrt[3]{(-8)^2} = \sqrt[3]{64} = 4

13. (x+y)^3 \cdot (x+y)^4 = (x+y)^{3+4} = (x+y)^7

Logarithms Practice

Translate into log notation:

1. 2^7 = 128 \Rightarrow \log_2 128 = 7

2. 9^{\frac{3}{2}} = 27 \Rightarrow \log_9 27 = \frac{3}{2}

3. 2^{-3} = \frac{1}{8} \Rightarrow \log_2 \frac{1}{8} = -3

Translate into exponential notation:

4. \log_3 9 = 2 \Rightarrow 3^2 = 9

5. \log_5 \frac{1}{5} = -1 \Rightarrow 5^{-1} = \frac{1}{5}

6. \log_{35} 1 = 0 \Rightarrow 35^0 = 1

Find N:

7. \log_{10} N = 2 \Rightarrow N = 10^2 = 100

8. \log_5 N = -2 \Rightarrow N = 5^{-2} = \frac{1}{25}

9. \log_{100} N = -\frac{7}{2} \Rightarrow N = 100^{-\frac{7}{2}} = (10^2)^{-\frac{7}{2}} = 10^{-7} = \frac{1}{10^7}

Solve for x:

10. \log_8 64 = x \Rightarrow 8^x = 64 = 8^2 \Rightarrow x = 2

11. \log_2 (\frac{1}{32}) = x \Rightarrow 2^x = \frac{1}{32} = 2^{-5} \Rightarrow x = -5

12. \log_9 \frac{1}{27} = x \Rightarrow 9^x = \frac{1}{27} \Rightarrow (3^2)^x = 3^{-3} \Rightarrow 3^{2x} = 3^{-3} \Rightarrow x = -\frac{3}{2}

Find a:

13. \log_a 64 = 3 \Rightarrow a^3 = 64 \Rightarrow a = \sqrt[3]{64} = 4

14. \log_a \frac{1}{7} = -2 \Rightarrow a^{-2} = \frac{1}{7} \Rightarrow \frac{1}{a^2} = \frac{1}{7} \Rightarrow a^2 = 7 \Rightarrow a = \sqrt{7}

15. \log_a 125 = \frac{3}{2} \Rightarrow a^{\frac{3}{2}} = 125 \Rightarrow a = 25

Given the log values below, find the numerical value of the following logarithms, using log properties:

\log 2 = 0.301, \quad \log 3 = 0.477, \quad \log 5 = 0.699, \quad \log 7 = 0.845
\ln 2 = 0.693, \quad \ln 3 = 1.099, \quad \ln 5 = 1.609, \quad \ln 7 = 1.946

16. \log 14 = \log (2 \cdot 7) = \log 2 + \log 7 = 0.301 + 0.845 = 1.146

17. \ln 8 = \ln (2^3) = 3 \ln 2 = 3(0.693) = 2.079

18. \log (\frac{5}{7}) = \log 5 - \log 7 = 0.699 - 0.845 = -0.146

19. \ln (\frac{5}{e}) = \ln 5 - \ln e = 1.609 - 1 = 0.609

20. \ln e^{100} = 100

21. \log_5 5^{102} = 102

Write as the sum or difference of simpler log quantities:

22. \ln (\frac{a^2 b^4}{z^3}) = 2 \ln a + 4 \ln b - 3 \ln z

Express as a single log with leading coefficient 1:

23. \frac{1}{2}(3 \ln x + \ln y - \ln z) = \frac{1}{2} (\ln x^3 + \ln y - \ln z) = \frac{1}{2} (\ln (x^3y) - \ln z) = \frac{1}{2} \ln (\frac{x^3y}{z}) = \ln \sqrt{\frac{x^3y}{z}}

Solve the following log equations:

24. \log 10 + \log 3 = \log x \Rightarrow \log (10 \cdot 3) = \log x \Rightarrow \log 30 = \log x \Rightarrow x = 30

25. \log (x+1) - \log (x-1) = \log 8 \Rightarrow \log (\frac{x+1}{x-1}) = \log 8 \Rightarrow \frac{x+1}{x-1} = 8 \Rightarrow x+1 = 8(x-1) \Rightarrow x+1 = 8x - 8 \Rightarrow 7x = 9 \Rightarrow x = \frac{9}{7}

26. \log2 x + \log2 (x+2) = 2 \Rightarrow \log_2 (x(x+2)) = 2 \Rightarrow x(x+2) = 2^2 \Rightarrow x^2 + 2x = 4 \Rightarrow x^2 + 2x - 4 = 0 \Rightarrow x = -1 + \sqrt{5}

Factoring Practice

Factor completely:

1. 2ax + 4a^2x^2 = 2ax(1 + 2ax)

2. x^3 + x^2 + x + 1 = x^2(x+1) + 1(x+1) = (x^2 + 1)(x+1)

3. 400 - 25y^2 = 25(16 - y^2) = 25(4-y)(4+y)

4. (a+b)^2 - c^2 = [(a+b)+c][(a+b)-c] = (a+b+c)(a+b-c)

5. x^6 - 7x^3 - 8 = (x^3)^2 - 7x^3 - 8 = (x^3 - 8)(x^3 + 1) = (x-2)(x^2+2x+4)(x+1)(x^2-x+1)

6. x^2 + 100 does not factor

7. 16m^2 - 56mn + 49n^2 = (4m - 7n)^2

8. 64 – 27x^3 = 4^3 - (3x)^3 = (4 - 3x)(16 + 12x + 9x^2)

9. x^2 + 3x + wx + 3w = x(x+3) + w(x+3) = (x+w)(x+3)

10. 1 - x^2 - 2xy - y^2 = 1 - (x^2 + 2xy + y^2) = 1 - (x+y)^2 = [1-(x+y)][1+(x+y)] = (1-x-y)(1+x+y)

11. 5x^5 + 30x^3 + 45x = 5x(x^4 + 6x^2 + 9) = 5x(x^2+3)^2

Algebraic Fractions Practice

Simplify as completely as possible:

1. \frac{3x^2 – 75y^2}{3x^2 – 21xy + 30y^2} = \frac{3(x^2 - 25y^2)}{3(x^2 - 7xy + 10y^2)} = \frac{(x-5y)(x+5y)}{(x-5y)(x-2y)} = \frac{x+5y}{x-2y}

2. \frac{a^2 – b^2}{b – a} = \frac{(a-b)(a+b)}{-(a-b)} = -(a+b) = -a - b

3. \frac{4–2x}{6x+30} \cdot \frac{x^2–25}{x^2 –7x+10} = \frac{-2(x-2)}{6(x+5)} \cdot \frac{(x-5)(x+5)}{(x-5)(x-2)} = \frac{-2}{6} = -\frac{1}{3}

4. \frac{5x^2 –9x–2}{30x^3+6x^2} \div \frac{x^4–3x^2–4}{2x^8 +6x^7+4x^6} = \frac{5x^2 –9x–2}{30x^3+6x^2} \cdot \frac{2x^8 +6x^7+4x^6}{x^4–3x^2–4} = \frac{(5x+1)(x-2)}{6x^2(5x+1)} \cdot \frac{2x^6(x^2+3x+2)}{(x^2-4)(x^2+1)} = \frac{(x-2)}{6x^2} \cdot \frac{2x^6(x+1)(x+2)}{(x-2)(x+2)(x^2+1)} = \frac{x^4(x+1)}{3(x^2+1)}

5. (\frac{1– x}{y}) \div (\frac{y – x^2}{y}) = (\frac{1– x}{y}) \cdot (\frac{y}{y – x^2}) = \frac{1– x}{y-x^2} = \frac{1-x}{(y-x)(1+x)} = \frac{-(x-1)}{(y-x)(1+x)} = \frac{-(x-1)}{1+y}

6. \frac{5}{x+2} + \frac{5}{2–x} – \frac{6}{x^2–4} = \frac{5}{x+2} - \frac{5}{x-2} - \frac{6}{(x-2)(x+2)}= \frac{5(x-2) - 5(x+2) - 6}{(x-2)(x+2)} = \frac{5x - 10 -5x - 10 - 6}{(x-2)(x+2)} = \frac{-26}{(x-2)(x+2)} = \frac{-26}{(4-x^2)} = \frac{26}{(4-x^2)}

7. \frac{3x–4}{x^2 —9} – \frac{2x–3}{x^2 –x–6} = \frac{3x-4}{(x-3)(x+3)} - \frac{2x-3}{(x-3)(x+2)} = \frac{(3x-4)(x+2)-(2x-3)(x+3)}{(x-3)(x+3)(x+2)} = \frac{3x^2+6x-4x-8-(2x^2+6x-3x-9)}{(x-3)(x+3)(x+2)} = \frac{x^2–x+1}{(x+3)(x–3)(x+2)}

Equations Practice

Solve for x:

1. 3(x - 4) + 5(x+6) = 14 \Rightarrow 3x - 12 + 5x + 30 = 14 \Rightarrow 8x + 18 = 14 \Rightarrow 8x = -4 \Rightarrow x = -\frac{1}{2}

2. \frac{x–4}{2} – \frac{x}{5} = \frac{1}{10} \Rightarrow \frac{5(x-4) - 2x}{10} = \frac{1}{10} \Rightarrow 5x -20 - 2x = 1 \Rightarrow 3x = 21 \Rightarrow x = 7

3. \frac{4}{x+1} = \frac{3}{x} + \frac{1}{15} \Rightarrow \frac{4}{x+1} = \frac{45+x}{15x} \Rightarrow 60x=(45+x)(x+1) \Rightarrow x=5,9

4. \frac{6–x}{x^2–4} – 2 = \frac{x}{x+2} \Rightarrow \frac{6-x}{(x-2)(x+2)}-\frac{x}{x+2}=2

5. x^2 – 8x = –10 \Rightarrow x^2 - 8x + 10 = 0 \Rightarrow x = \frac{8 \pm \sqrt{64 - 40}}{2} = \frac{8 \pm \sqrt{24}}{2} = \frac{8 \pm 2\sqrt{6}}{2} = 4 \pm \sqrt{6}

6. 3x^2 – x– 2 = 0 \Rightarrow (3x + 2)(x - 1) = 0 \Rightarrow x = 1 \text{ or } x = -\frac{2}{3}

Inequalities Practice

Solve the following inequalities. Give your answer in both inequality and interval notation.

1. 3x + 2 \geq 0 \Rightarrow 3x \geq -2 \Rightarrow x \geq -\frac{2}{3}; [-\frac{2}{3}, \infty)

2. 7 – 4x < 5 \Rightarrow – 4x < -2 \Rightarrow x > \frac{1}{2}; (\frac{1}{2}, \infty)

3. |2x + 1| < 2 \Rightarrow -2<2x+1<2\Rightarrow -3<2x<1 \Rightarrow -3/2 < x < 1/2; (-3/2, 1/2)

4. |4 – 3x| \geq 5 \Rightarrow (4-3x)\leq -5 or (4-3x)\geq 5 \Rightarrow -3x\leq -9 or -3x\geq 1 \Rightarrow x \geq 3 or x\leq -1/3; (-inf, -\frac{1}{3}] \cup [3, inf)

5. $$x^2 + 2x – 3 < 0 \Rightarrow (x+3)(x-1)<0 \Rightarrow -3