USTET Mathematics — Algebra — Sets, Exponents, Radicals, Polynomials & EquationsFlash Cards

Find A ∪ B if A = {1, 3, 5, 7} and B = {2, 4, 6, 7}

Step 1: Union means all elements in either A or B (no duplicates). Step 2: List all elements from A: 1, 3, 5, 7. Step 3: Add elements from B not already in A: 2, 4, 6. Step 4: Note that 7 appears in both, so write it only once. Answer: A ∪ B = {1, 2, 3, 4, 5, 6, 7}

Find A ∩ B if A = {2, 4, 6, 8, 10} and B = {4, 8, 12, 16}

Step 1: Intersection means elements common to both A and B. Step 2: Check each element in A: 2 (not in B), 4 (in B ✓), 6 (not in B), 8 (in B ✓), 10 (not in B). Step 3: List common elements. Answer: A ∩ B = {4, 8}

Simplify: 2³ × 2⁵

Step 1: Apply product rule for exponents: aᵐ × aⁿ = aᵐ⁺ⁿ. Step 2: Same base (2), so add exponents: 2³ × 2⁵ = 2³⁺⁵. Step 3: Calculate: 2³⁺⁵ = 2⁸. Step 4: Evaluate: 2⁸ = 256. Answer: 256

Simplify: (3x²)³

Step 1: Apply power rule: (aᵐ)ⁿ = aᵐⁿ and (ab)ⁿ = aⁿbⁿ. Step 2: Raise each factor to the 3rd power: (3x²)³ = 3³ × (x²)³. Step 3: Calculate: 3³ = 27 and (x²)³ = x²ˣ³ = x⁶. Step 4: Combine results. Answer: 27x⁶

Solve for x: 2ˣ = 32

Step 1: Express 32 as a power of 2: 32 = 2⁵. Step 2: Substitute: 2ˣ = 2⁵. Step 3: Since bases are equal, exponents must be equal: x = 5. Step 4: Verify: 2⁵ = 32 ✓. Answer: x = 5

Simplify: √48

Step 1: Factor 48 into perfect square factors: 48 = 16 × 3 = 4² × 3. Step 2: Apply radical rule: √(a×b) = √a × √b. Step 3: √48 = √(16×3) = √16 × √3. Step 4: Simplify perfect square: √16 = 4. Answer: 4√3

Simplify: ∛54

Step 1: Factor 54 into perfect cube factors: 54 = 27 × 2 = 3³ × 2. Step 2: Apply cube root rule: ∛(a×b) = ∛a × ∛b. Step 3: ∛54 = ∛(27×2) = ∛27 × ∛2. Step 4: Simplify perfect cube: ∛27 = 3. Answer: 3∛2

Rationalize the denominator: 6/√3

Step 1: To eliminate √3 from denominator, multiply by √3/√3. Step 2: (6/√3) × (√3/√3) = (6√3)/(√3×√3). Step 3: Simplify denominator: √3×√3 = 3. Step 4: Result: 6√3/3. Step 5: Simplify fraction: 6√3/3 = 2√3. Answer: 2√3

Add: (3x² + 5x - 2) + (2x² - 3x + 7)

Step 1: Remove parentheses: 3x² + 5x - 2 + 2x² - 3x + 7. Step 2: Group like terms: (3x² + 2x²) + (5x - 3x) + (-2 + 7). Step 3: Combine like terms: 5x² + 2x + 5. Answer: 5x² + 2x + 5

Subtract: (4x³ - 2x + 1) - (x³ + 5x - 3)

Step 1: Distribute negative sign: 4x³ - 2x + 1 - x³ - 5x + 3. Step 2: Group like terms: (4x³ - x³) + (-2x - 5x) + (1 + 3). Step 3: Combine like terms: 3x³ - 7x + 4. Answer: 3x³ - 7x + 4

Multiply: (x + 4)(x - 3)

Step 1: Use FOIL method (First, Outer, Inner, Last). Step 2: First: x × x = x². Step 3: Outer: x × (-3) = -3x. Step 4: Inner: 4 × x = 4x. Step 5: Last: 4 × (-3) = -12. Step 6: Combine: x² - 3x + 4x - 12 = x² + x - 12. Answer: x² + x - 12

Factor: x² + 7x + 12

Step 1: Find two numbers that multiply to 12 and add to 7. Step 2: List factor pairs of 12: (1,12), (2,6), (3,4). Step 3: Check sums: 1+12=13, 2+6=8, 3+4=7 ✓. Step 4: Use 3 and 4: x² + 7x + 12 = (x + 3)(x + 4). Step 5: Verify: (x+3)(x+4) = x² + 4x + 3x + 12 = x² + 7x + 12 ✓. Answer: (x + 3)(x + 4)

Factor: x² - 9

Step 1: Recognize difference of squares pattern: a² - b². Step 2: Identify: x² - 9 = x² - 3². Step 3: Apply formula: a² - b² = (a + b)(a - b). Step 4: Substitute: x² - 3² = (x + 3)(x - 3). Step 5: Verify: (x+3)(x-3) = x² - 3x + 3x - 9 = x² - 9 ✓. Answer: (x + 3)(x - 3)

Solve: 3x + 7 = 22

Step 1: Subtract 7 from both sides: 3x + 7 - 7 = 22 - 7. Step 2: Simplify: 3x = 15. Step 3: Divide both sides by 3: 3x/3 = 15/3. Step 4: Simplify: x = 5. Step 5: Check: 3(5) + 7 = 15 + 7 = 22 ✓. Answer: x = 5

Solve: 2(x - 3) = x + 4

Step 1: Distribute: 2x - 6 = x + 4. Step 2: Subtract x from both sides: 2x - x - 6 = x - x + 4. Step 3: Simplify: x - 6 = 4. Step 4: Add 6 to both sides: x - 6 + 6 = 4 + 6. Step 5: Simplify: x = 10. Step 6: Check: 2(10-3) = 2(7) = 14, and 10+4 = 14 ✓. Answer: x = 10

Solve using quadratic formula: x² - 5x + 6 = 0

Step 1: Identify a=1, b=-5, c=6. Step 2: Apply formula: x = [-b ± √(b²-4ac)] / (2a). Step 3: Calculate discriminant: b²-4ac = (-5)²-4(1)(6) = 25-24 = 1. Step 4: Substitute: x = [5 ± √1] / 2 = [5 ± 1] / 2. Step 5: Find solutions: x = (5+1)/2 = 3 or x = (5-1)/2 = 2. Answer: x = 2 or x = 3

Solve by factoring: x² - 4x - 5 = 0

Step 1: Find two numbers that multiply to -5 and add to -4. Step 2: Consider factor pairs: (-1,5), (1,-5). Step 3: Check: -1+5=4 (wrong sign), 1+(-5)=-4 ✓. Step 4: Factor: x² - 4x - 5 = (x + 1)(x - 5). Step 5: Set each factor to zero: x+1=0 or x-5=0. Step 6: Solve: x=-1 or x=5. Answer: x = -1 or x = 5

When do you use the quadratic formula instead of factoring?

Use quadratic formula when: 1) Trinomial doesn't factor easily with integers, 2) Discriminant is not a perfect square, 3) Coefficients are large or decimal. Example: x² + 3x + 1 = 0 has discriminant 9-4=5 (not perfect square), so factoring is difficult. Formula gives x = [-3 ± √5]/2.

Solve the system: x + y = 8, 2x - y = 1

Step 1: Use elimination method. Step 2: Add equations to eliminate y: (x+y) + (2x-y) = 8 + 1. Step 3: Simplify: 3x = 9, so x = 3. Step 4: Substitute x=3 into first equation: 3 + y = 8. Step 5: Solve for y: y = 5. Step 6: Check in both equations: 3+5=8 ✓, 2(3)-5=1 ✓. Answer: x = 3, y = 5

Express in scientific notation: 0.00045

Step 1: Move decimal point right until you have one non-zero digit before decimal. Step 2: Count moves: 0.00045 → 4.5 × 10⁻⁴ (moved 4 places right). Step 3: Since we moved right, exponent is negative. Step 4: Verify: 4.5 × 10⁻⁴ = 4.5 × 0.0001 = 0.00045 ✓. Answer: 4.5 × 10⁻⁴