This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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QUESTION 2 2.4. Step 1: Evaluate the cube root of 8. [3]8 = 2 Step 2: Square the result. (2)^2 = 4 The simplified answer is 4. 2.5. Step 1: Evaluate (-3)^2. (-3)^2 = 9 Step 2: Evaluate (-5)^2. (-5)^2 = 25 Step 3: Multiply the results. 9 × 25 = 225 The simplified answer is 225. 2.6. Step 1: Convert the mixed number to a decimal. -5 (91)/(100) = -5.91 The decimal form is -5.91. 2.7. Step 1: Convert the mixed number to an improper fraction. 3(5)/(7) = (3 × 7 + 5)/(7) = (26)/(7) Step 2: Find a common denominator for (3)/(4) and (26)/(7), which is 28. (3)/(4) + (26)/(7) = (3 × 7)/(4 × 7) + (26 × 4)/(7 × 4) = (21)/(28) + (104)/(28) Step 3: Add the fractions. (21 + 104)/(28) = (125)/(28) The simplified answer is (125)/(28). 2.8. Step 1: Multiply the numerators and the denominators. ((2)/(3))((1)/(2))(-(1)/(5)) = (2 × 1 × (-1))/(3 × 2 × 5) = (-2)/(30) Step 2: Simplify the fraction. (-2)/(30) = -(1)/(15) The simplified answer is -(1)/(15). 2.9. Step 1: To divide by a fraction, multiply by its reciprocal. (3)/(5) ÷ (2)/(7) = (3)/(5) × (7)/(2) Step 2: Multiply the numerators and the denominators. (3 × 7)/(5 × 2) = (21)/(10) The simplified answer is (21)/(10). 2.10. Step 1: Divide the coefficients. (12)/(4) = 3 Step 2: Divide the variables using the exponent rule (a^m)/(a^n) = a^m-n. (x^4)/(x^3) = x^4-3 = x^1 = x Step 3: Combine the results. 3x The simplified answer is 3x. 2.11. Step 1: Apply the power rules (xy)^n = x^n y^n and (x^m)^n = x^mn. (3b^2)^3 = 3^3 (b^2)^3 = 27b^6 (a^3)^2 = a^3 × 2 = a^6 Step 2: Multiply the simplified terms. 27b^6 × a^6 = 27a^6b^6 The simplified answer is 27a^6b^6. 2.12. Step 1: Perform the multiplications. 4x × x = 4x^2 5x × x = 5x^2 Step 2: Subtract the like terms. 4x^2 - 5x^2 = -x^2 The simplified answer is -x^2. 2.13. Step 1: Combine the like terms inside the parentheses. (2x + 5x) = 7x Step 2: Multiply the result by 3. 7x × 3 = 21x The simplified answer is 21x. 2.14. Step 1: Combine the like terms inside the parentheses. (3x^2 - 2x^2) = x^2 Step 2: Multiply 4x by the result. 4x × x^2 = 4x^1+2 = 4x^3 The simplified answer is 4x^3. 2.15. Step 1: Multiply the coefficients and variables. (-3x^2y)(-3xy^2)(-3x^2y)(-3xy^2) = ((-3) × (-3) × (-3) × (-3)) × (x^2 · x · x^2 · x) × (y · y^2 · y · y^2) = 81 × x^2+1+2+1 × y^1+2+1+2 = 81x^6y^6 The simplified answer is 81x^6y^6. 2.16. Step 1: Factor out the common term from the numerator. (7x^3b + 14ab^2)/(7ab) = (7b(x^3 + 2ab))/(7ab) Step 2: Cancel out the common terms in the numerator and denominator. (7b(x^3 + 2ab))/(7ab) = (x^3 + 2ab)/(a) The simplified answer is (x^3 + 2ab)/(a). 2.17. Step 1: Subtract 4 from both sides of the equation. 4 + 2x = 12 2x = 12 - 4 2x = 8 Step 2: Divide both sides by 2. x = (8)/(2) x = 4 The solution is x=4. 2.18. Step 1: Express 27 as a power of 3. 3^2x+2 = 3^3 Step 2: Equate the exponents. 2x+2 = 3 Step 3: Subtract 2 from both sides. 2x = 3 - 2 2x = 1 Step 4: Divide by 2. x = (1)/(2) The solution is x=(1)/(2). 2.19. Step 1: Distribute the -3 on the left side. 4x - 3(x-4) = 3 - 2x 4x - 3x + 12 = 3 - 2x Step 2: Combine like terms on the left side. x + 12 = 3 - 2x Step 3: Add 2x to both sides. x + 2x + 12 = 3 3x + 12 = 3 Step 4: Subtract 12 from both sides. 3x = 3 - 12 3x = -9 Step 5: Divide by 3. x = (-9)/(3) x = -3 The solution is x=-3. 2.20. Step 1: Add the decimals. 0.05 + 0.1 = 0.15 Step 2: Convert the decimal to a fraction. 0.15 = (15)/(100) Step 3: Simplify the fraction. (15 ÷ 5)/(100 ÷ 5) = (3)/(20) The answer as a common fraction is (3)/(20). 2.21. Step 1: Calculate the cube of -0.5. (-0.5)^3 = (-0.5) × (-0.5) × (-0.5) = 0.25 × (-0.5) = -0.125 The answer rounded to three decimal places is -0.125. QUESTION 3 3.1. Consider the following pattern -7; -11; -15; ... 3.1.1. The rule is to subtract 4 from the previous term. 3.1.2. Step 1: Calculate the difference between consecutive terms. -11 - (-7) = -4 -15 - (-11) = -4 The common difference is -4. 3.1.3. Step 1: Add the common difference to the last given term. -15 + (-4) = -19 Step 2: Add the common difference to the new term. -19 + (-4) = -23 The next two terms are -19; -23. 3.1.4. Step 1: Use the formula for the n-th term of an arithmetic sequence, T_n = a + (n-1)d, where a = -7 and d = -4. T_n = -7 + (n-1)(-4) Step 2: Simplify the expression. T_n = -7 - 4n + 4 T_n = -4n - 3 The rule is T_n = -4n - 3. 3.1.5. Step 1: Substitute n=30 into the formula T_n = -4n - 3. T_30 = -4(30) - 3 Step 2: Calculate the value. T_30 = -120 - 3 T_30 = -123 The 30th term is -123. 3.2. Step 1: The exponent -7 means move the decimal point 7 places to the left. 1.23 × 10^-7 = 0.000000123 The number in full is 0.000000123. 3.3. This question is incomplete. Please provide the full question. That's 2 down. 3 left today — send the next one.