Comprehensive Study Guide: Algebra, Surds, Indices, Inequalities, and Modulus

The following tables provide essential numerical references for rapid calculation.
Squares of Integers from 1 to 30:
- $1^2 = 1$
- $2^2 = 4$
- $3^2 = 9$
- $4^2 = 16$
- $5^2 = 25$
- $6^2 = 36$
- $7^2 = 49$
- $8^2 = 64$
- $9^2 = 81$
- $10^2 = 100$
- $11^2 = 121$
- $12^2 = 144$
- $13^2 = 169$
- $14^2 = 196$
- $15^2 = 225$
- $16^2 = 256$
- $17^2 = 289$
- $18^2 = 324$
- $19^2 = 361$
- $20^2 = 400$
- $21^2 = 441$
- $22^2 = 484$
- $23^2 = 529$
- $24^2 = 576$
- $25^2 = 625$
- $26^2 = 676$
- $27^2 = 729$
- $28^2 = 784$
- $29^2 = 841$
- $30^2 = 900$
Cubes of Integers from 2 to 9:
- $2^3 = 8$
- $3^3 = 27$
- $4^3 = 64$
- $5^3 = 125$
- $6^3 = 216$
- $7^3 = 343$
- $8^3 = 512$
- $9^3 = 729$
Fundamental Square Roots:
- $\sqrt{2} \approx 1.41$
- $\sqrt{3} \approx 1.73$
- $\sqrt{5} \approx 2.23$

Halves and Quarters:
- $\frac{1}{2} = 0.5$
- $\frac{1}{4} = 0.25$
- $\frac{2}{4} = 0.5$
- $\frac{3}{4} = 0.75$
Thirds and Sixths:
- $\frac{1}{3} = 0.333$
- $\frac{2}{3} = 0.667$
- $\frac{1}{6} = 0.1667$
- $\frac{2}{6} = 0.3333$
- $\frac{4}{6} = 0.6667$
- $\frac{5}{6} = 0.8333$
Fifths:
- $\frac{1}{5} = 0.2$
- $\frac{2}{5} = 0.4$
- $\frac{3}{5} = 0.6$
- $\frac{4}{5} = 0.8$
Sevenths (Approximate values):
- $\frac{1}{7} \approx 0.143$
- $\frac{2}{7} \approx 0.286$
- $\frac{3}{7} \approx 0.429$
- $\frac{4}{7} \approx 0.571$
- $\frac{5}{7} \approx 0.714$
- $\frac{6}{7} \approx 0.857$
Eights:
- $\frac{1}{8} = 0.125$
- $\frac{2}{8} = 0.25$
- $\frac{3}{8} = 0.375$
- $\frac{4}{8} = 0.5$
- $\frac{5}{8} = 0.625$
- $\frac{6}{8} = 0.75$
- $\frac{7}{8} = 0.875$
Ninths:
- $\frac{1}{9} = 0.1111$
- $\frac{2}{9} = 0.2222$
- $\frac{3}{9} = 0.3333$
- $\frac{4}{9} = 0.4444$
- $\frac{5}{9} = 0.5556$
- $\frac{6}{9} = 0.6667$
- $\frac{7}{9} = 0.7778$
- $\frac{8}{9} = 0.8889$

Grouping Terms and Signed Numbers:
- Example: $(19 - 18 - 17 - 16) - (20 - 19 - 18 - 17)$ . Simplified: $(-32) - (-34) = -32 + 34 = 2$ .
Place Value and Expanded Form:
- Example: $(3 \times 100) + (4 \times 1) + (5 \times 1,000) + (6 \times 10)$ . Standard form calculation: $300 + 4 + 5000 + 60 = 5364$ .
Factoring and Distributive Properties:
- Identifying equivalent expressions: $456(72) + 28(456)$ is equal to $(456)(72 + 28) = 456(100) = 45,600$ .
Division and Ratios:
- Calculation of $3204 \div 160 = 20.025$ .
- Division of fractions: $\frac{5}{12} \div \frac{5}{9} = \frac{5}{12} \times \frac{9}{5} = \frac{9}{12} = \frac{3}{4}$ .
Evaluating Decimal Expressions:
- Division of decimals: $\frac{0.707}{0.7} = 1.01$ .
- Multiplication of decimals: $54.36 \times 0.12 = 6.5232$ .
- Addition of large numbers: $5661 + 7110 + 5011 = 17782$ .
- Subtraction: $900,000 - 640,001 = 259,999$ .

Formula Rearrangement:
- Solving $z = 5x - 25xy$ for $x$ :
  - $z = x(5 - 25y)$
  - $x = \frac{z}{5 - 25y}$
- Solving distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ for $y_2$ :
  - $d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$
  - $d^2 - (x_2 - x_1)^2 = (y_2 - y_1)^2$
  - $\sqrt{d^2 - (x_2 - x_1)^2} = y_2 - y_1$
  - $y_2 = \sqrt{d^2 - (x_2 - x_1)^2} + y_1$
- Solving $p = \frac{5b}{c^2}$ for $c$ :
  - Given $p = 9$ and $b = 20$
  - $9 = \frac{5 \times 20}{c^2}$
  - $9 = \frac{100}{c^2} \rightarrow 9c^2 = 100$
  - $c^2 = \frac{100}{9} \rightarrow c = \frac{10}{3}$ .
Bakery Business Scenario:
- A bakery orders vanilla beans at $\$12.45$ per package of 10, with a flat shipping cost of $\$6.00$ .
- Let $c$ be the total cost and $p$ be the number of packages.
- Equation: $c = 12.45p + 6.00$
- Solved for $p$ : $p = \frac{c - 6}{12.45}$ .

Standard Expansions:
- Binomial Square (Sum): $(a + b)^2 = a^2 + 2ab + b^2$ .
- Binomial Square (Difference): $(a - b)^2 = a^2 - 2ab + b^2$ .
- Difference of Squares: $(a + b)(a - b) = a^2 - b^2$ .
Factoring Quadratics:
- Factor $x^2 - 16x + 64$ : $(x - 8)^2$ .
- Factor $x^2 - 2x - 15$ : Solutions are $x = 5$ and $x = -3$ .
- Factor $4x^2 + 4xy + y^2$ : $(2x + y)^2$ .
Evaluating Identities with Given Values:
- If $(p - q)^2 = 25$ and $pq = 14$ , find $(p + q)^2$ :
  - $(p + q)^2 = (p - q)^2 + 4pq = 25 + 4(14) = 25 + 56 = 81$ .
- If $(a + b)^2 = 9$ and $(a - b)^2 = 49$ , find $a^2 + b^2$ :
  - $(a + b)^2 + (a - b)^2 = 2(a^2 + b^2)$
  - $9 + 49 = 58$
  - $a^2 + b^2 = \frac{58}{2} = 29$ .
Higher Power Factoring:
- $x^4 - 25 = (x^2 - 5)(x^2 + 5)$ .
- $(x + 2)(x^4 + 16)(x - 2)(x^2 + 4)$ . This represents a chain of difference of squares:
  1. $(x + 2)(x - 2) = x^2 - 4$
  2. $(x^2 - 4)(x^2 + 4) = x^4 - 16$
  3. $(x^4 - 16)(x^4 + 16) = x^8 - 256$ .

Laws of Exponents:
- Multiplicative Law: $p \times q \times p \times q^8 = p^2 q^9$ .
- Power of a Power: $(x^2 y^3)^4 = x^8 y^{12}$ .
- Division of Powers: $\frac{x^6 y^6}{x^4 y^2} = x^2 y^4$ .
- Negative and Fractional Exponents: $(0.04)^{-1.5} = (\frac{4}{100})^{-\frac{3}{2}} = (\frac{1}{25})^{-\frac{3}{2}} = 25^{\frac{3}{2}} = (\sqrt{25})^3 = 5^3 = 125$ .
Properties of Surds Root Operations:
- Multiplication: $\sqrt{18x} \times \sqrt{2x} = \sqrt{36x^2} = 6x$ .
- Addition/Subtraction: $5\sqrt{162} - 3\sqrt{18} - 4\sqrt{98}$ .
  - Simplify terms: $5(\sqrt{81 \times 2}) - 3(\sqrt{9 \times 2}) - 4(\sqrt{49 \times 2})$
  - $5(9\sqrt{2}) - 3(3\sqrt{2}) - 4(7\sqrt{2})$
  - $45\sqrt{2} - 9\sqrt{2} - 28\sqrt{2} = 8\sqrt{2}$ .
- Sum of radicals: $\sqrt{12} + \sqrt{27} = 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}$ .
- Square of binomial surds: $(\sqrt{3} - \sqrt{8})^2 = (\sqrt{3})^2 - 2\sqrt{3}\sqrt{8} + (\sqrt{8})^2 = 3 - 2\sqrt{24} + 8 = 11 - 2(2\sqrt{6}) = 11 - 4\sqrt{6}$ .
Complex Radical Equations:
- Solving $8\sqrt{x + 7} = 3\sqrt{x + 17}$ :
  - $64(x + 7) = 9(x + 17)$
  - $64x + 448 = 9x + 153$
  - $55x = -295$
  - $x = -\frac{295}{55} = -\frac{59}{11}$ .

Linear Inequalities:
- Example 1: $2x - 5 < 5 \rightarrow 2x < 10 \rightarrow x < 5$ .
- Example 2: $-4x - 12 < 15 \rightarrow -4x < 27 \rightarrow x > -\frac{27}{4}$ .
- Example 3: $10 - 3x < 2x - 5 \rightarrow 15 < 5x \rightarrow 3 < x \text{ or } x > 3$ .
Compound Inequalities:
- Solve $-3 < 2x + 5 < 9$ :
  - $-8 < 2x < 4$
  - $-4 < x < 2$ .
Quadratic Inequalities:
- Solve $x^2 - 36 > 0$
  - $(x - 6)(x + 6) > 0$
  - Possible values: $x < -6$ or $x > 6$ .
- Solve $5x + x^2 + 6 < 0$
  - $(x + 3)(x + 2) < 0$
  - Solution: $-3 < x < -2$ .
Integer Solutions:
- Find the largest integer $p$ for $4 + 3p < p + 1$ :
  - $2p < -3 \rightarrow p < -1.5$
  - Largest integer is $-2$ .

Fundamental Modulus Equations:
- Solve $|3x - 11| = 8$ :
  - $3x - 11 = 8 \rightarrow 3x = 19 \rightarrow x = \frac{19}{3}$
  - $3x - 11 = -8 \rightarrow 3x = 3 \rightarrow x = 1$ .
- Solve $|x - 2| = |2x + 1|$ :
  - Case 1: $x - 2 = 2x + 1 \rightarrow -3 = x$
  - Case 2: $x - 2 = -(2x + 1) \rightarrow x - 2 = -2x - 1 \rightarrow 3x = 1 \rightarrow x = \frac{1}{3}$ .
Modulus Inequalities:
- Solve $|1 - 2x| < 7$ :
  - $-7 < 1 - 2x < 7$
  - $-8 < -2x < 6$
  - Divide by $-2$ (flip inequality): $4 > x > -3$ . Range: $-3 < x < 4$ .
- Solve for constant range: High-temperature control in a greenhouse.
  - Temperature $F$ does not vary from $79^{\circ}$ by more than $7^{\circ}$ .
  - Inequality: $|F - 79| \leq 7$ .
Geometric Application of Modulus:
- The area bounded by the relation $|x| + |y| = 4$ :
  - This forms a square with vertices at $(4,0), (0,4), (-4,0), \text{ and } (0,-4)$ .
  - Side length $s = \sqrt{4^2 + 4^2} = \sqrt{32}$ .
  - Area = $s^2 = 32$ .