Simplify the following expressions and solve the inequalities: a) y2 y3, b) 50 6, c) (F6 F3) / (F4 F2), d) n7 / n2, e) 56 / 54, f) 9c5 / 9c2, and g) 1/4x - 4 <= 6.

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Asked on April 24, 2026Mathematics

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.

Accepted Answer · 2026-04-24T13:12:48.414Z

Here are the solutions to your assignment questions. 1. Simplify a) y^2 × y^3 Step 1: Apply the rule a^m × a^n = a^m+n. y^2 × y^3 = y^2+3 = y^5 The simplified expression is y^5. b) 5^0 × 6 Step 1: Apply the rule a^0 = 1. 5^0 × 6 = 1 × 6 = 6 The simplified expression is 6. c) (F^6 × F^3) ÷ (F^4 × F^2) Step 1: Simplify the terms in the parentheses using a^m × a^n = a^m+n. (F^6 × F^3) = F^6+3 = F^9 (F^4 × F^2) = F^4+2 = F^6 Step 2: Perform the division using a^m ÷ a^n = a^m-n. F^9 ÷ F^6 = F^9-6 = F^3 The simplified expression is F^3. d) n^7 ÷ n^2 Step 1: Apply the rule a^m ÷ a^n = a^m-n. n^7 ÷ n^2 = n^7-2 = n^5 The simplified expression is n^5. e) 5^6 ÷ 5^4 Step 1: Apply the rule a^m ÷ a^n = a^m-n. 5^6 ÷ 5^4 = 5^6-4 = 5^2 Step 2: Calculate the value. 5^2 = 25 The simplified expression is 25. f) 9c^5 ÷ 9c^2 Step 1: Divide the numerical coefficients and apply the rule a^m ÷ a^n = a^m-n for the variables. 9c^5 ÷ 9c^2 = (9)/(9) × (c^5)/(c^2) = 1 × c^5-2 = c^3 The simplified expression is c^3. 2. Solve the following inequalities a) (1)/(4)x - 4 6 Step 1: Add 4 to both sides of the inequality. (1)/(4)x - 4 + 4 6 + 4 (1)/(4)x 10 Step 2: Multiply both sides by 4. 4 × (1)/(4)x 10 × 4 x 40 The solution is x 40. b) 3x + 5 > -4 Step 1: Subtract 5 from both sides of the inequality. 3x + 5 - 5 > -4 - 5 3x > -9 Step 2: Divide both sides by 3. (3x)/(3) > (-9)/(3) x > -3 The solution is x > -3. c) (3 - 2x)/(3) > -8(1)/(3) Step 1: Convert the mixed number to an improper fraction. -8(1)/(3) = -((8 × 3) + 1)/(3) = -(24+1)/(3) = -(25)/(3) The inequality becomes: (3 - 2x)/(3) > -(25)/(3) Step 2: Multiply both sides by 3. 3 × (3 - 2x)/(3) > 3 × -(25)/(3) 3 - 2x > -25 Step 3: Subtract 3 from both sides. 3 - 2x - 3 > -25 - 3 -2x > -28 Step 4: Divide both sides by -2. Remember to reverse the inequality sign when dividing by a negative number. (-2x)/(-2) < (-28)/(-2) x < 14 The solution is x < 14. d) 5(2y+9) 3(2y+9) Step 1: Expand both sides of the inequality. 10y + 45 6y + 27 Step 2: Subtract 6y from both sides. 10y - 6y + 45 6y - 6y + 27 4y + 45 27 Step 3: Subtract 45 from both sides. 4y + 45 - 45 27 - 45 4y -18 Step 4: Divide both sides by 4. (4y)/(4) (-18)/(4) y -(9)/(2) y -4.5 The solution is y -4.5. 3. Work out the x and y intercept for the line whose equations are a) y = 4x + 3 To find the x-intercept, set y=0: 0 = 4x + 3 -3 = 4x x = -(3)/(4) To find the y-intercept, set x=0: y = 4(0) + 3 y = 3 The x-intercept is (-(3)/(4), 0) and the y-intercept is (0, 3). b) 3x + 2y = 12 To find the x-intercept, set y=0: 3x + 2(0) = 12 3x = 12 x = (12)/(3) x = 4 To find the y-intercept, set x=0: 3(0) + 2y = 12 2y = 12 y = (12)/(2) y = 6 The x-intercept is (4, 0) and the y-intercept is (0, 6). c) y = -2x + 3 To find the x-intercept, set y=0: 0 = -2x + 3 2x = 3 x = (3)/(2) To find the y-intercept, set x=0: y = -2(0) + 3 y = 3 The x-intercept is ((3)/(2), 0) and the y-intercept is (0, 3). 4. Find the volume and surface of the following a) sphere radius 3.5 cm Volume of a sphere: V = (4)/(3) r^3 Surface area of a sphere: SA = 4 r^2 Given r = 3.5 cm. Step 1: Calculate the volume. V = (4)/(3) × × (3.5 cm)^3 V = (4)/(3) × × 42.875 cm^3 V ≈ 179.59 cm^3 Step 2: Calculate the surface area. SA = 4 × × (3.5 cm)^2 SA = 4 × × 12.25 cm^2 SA ≈ 153.94 cm^2 The volume is 179.59 cm^3 and the surface area is 153.94 cm^2. b) cone radius 7 cm and slant height 10 cm Volume of a cone: V = (1)/(3) r^2 h Surface area of a cone: SA = r (r+l) Given r = 7 cm and l = 10 cm. First, find the height h using the Pythagorean theorem: h = sqrt(l^2 - r^2). h = sqrt((10 cm))^2 - (7 cm)^2 h = sqrt(100 cm)^2 - 49 cm^2 h = sqrt(51) cm ≈ 7.14 cm Step 1: Calculate the volume. V = (1)/(3) × × (7 cm)^2 × sqrt(51) cm V = (1)/(3) × × 49 cm^2 × 7.1414 cm V ≈ 366.49 cm^3 Step 2: Calculate the surface area. SA = × 7 cm × (7 cm + 10 cm) SA = × 7 cm × 17 cm SA = 119 cm^2 SA ≈ 373.85 cm^2 The volume is 366.49 cm^3 and the surface area is 373.85 cm^2. c) cuboid of 5 m x 2 m x 3.2 m Volume of a cuboid: V = lwh Surface area of a cuboid: SA = 2(lw + lh + wh) Given l = 5 m, w = 2 m, h = 3.2 m. Step 1: Calculate the volume. V = 5 m × 2 m × 3.2 m V = 32 m^3 Step 2: Calculate the surface area. SA = 2((5 m × 2 m) + (5 m × 3.2 m) + (2 m × 3.2 m)) SA = 2(10 m^2 + 16 m^2 + 6.4 m^2) SA = 2(32.4 m^2) SA = 64.8 m^2 The volume is 32 m^3 and the surface area is 64.8 m^2. d) cube of sides 4 m Volume of a cube: V = s^3 Surface area of a cube: SA = 6s^2 Given s = 4 m. Step 1: Calculate the volume. V = (4 m)^3 V = 64 m^3 Step 2: Calculate the surface area. SA = 6 × (4 m)^2 SA = 6 × 16 m^2 SA = 96 m^2 The volume is 64 m^3 and the surface area is 96 m^2. 5. A car is brought to rest from 90 km/h in 20 seconds. What is its acceleration? Given: Initial velocity, v_i = 90 km/h Final velocity, v_f = 0 km/h (brought to rest) Time, t = 20 s Step 1: Convert initial velocity from km/

Accepted Answer · 2026-04-24T13:12:48.414Z

Here are the solutions to your assignment questions. 1. Simplify a) y^2 × y^3 Step 1: Apply the rule a^m × a^n = a^m+n. y^2 × y^3 = y^2+3 = y^5 The simplified expression is y^5. b) 5^0 × 6 Step 1: Apply the rule a^0 = 1. 5^0 × 6 = 1 × 6 = 6 The simplified expression is 6. c) (F^6 × F^3) ÷ (F^4 × F^2) Step 1: Simplify the terms in the parentheses using a^m × a^n = a^m+n. (F^6 × F^3) = F^6+3 = F^9 (F^4 × F^2) = F^4+2 = F^6 Step 2: Perform the division using a^m ÷ a^n = a^m-n. F^9 ÷ F^6 = F^9-6 = F^3 The simplified expression is F^3. d) n^7 ÷ n^2 Step 1: Apply the rule a^m ÷ a^n = a^m-n. n^7 ÷ n^2 = n^7-2 = n^5 The simplified expression is n^5. e) 5^6 ÷ 5^4 Step 1: Apply the rule a^m ÷ a^n = a^m-n. 5^6 ÷ 5^4 = 5^6-4 = 5^2 Step 2: Calculate the value. 5^2 = 25 The simplified expression is 25. f) 9c^5 ÷ 9c^2 Step 1: Divide the numerical coefficients and apply the rule a^m ÷ a^n = a^m-n for the variables. 9c^5 ÷ 9c^2 = (9)/(9) × (c^5)/(c^2) = 1 × c^5-2 = c^3 The simplified expression is c^3. 2. Solve the following inequalities a) (1)/(4)x - 4 6 Step 1: Add 4 to both sides of the inequality. (1)/(4)x - 4 + 4 6 + 4 (1)/(4)x 10 Step 2: Multiply both sides by 4. 4 × (1)/(4)x 10 × 4 x 40 The solution is x 40. b) 3x + 5 > -4 Step 1: Subtract 5 from both sides of the inequality. 3x + 5 - 5 > -4 - 5 3x > -9 Step 2: Divide both sides by 3. (3x)/(3) > (-9)/(3) x > -3 The solution is x > -3. c) (3 - 2x)/(3) > -8(1)/(3) Step 1: Convert the mixed number to an improper fraction. -8(1)/(3) = -((8 × 3) + 1)/(3) = -(24+1)/(3) = -(25)/(3) The inequality becomes: (3 - 2x)/(3) > -(25)/(3) Step 2: Multiply both sides by 3. 3 × (3 - 2x)/(3) > 3 × -(25)/(3) 3 - 2x > -25 Step 3: Subtract 3 from both sides. 3 - 2x - 3 > -25 - 3 -2x > -28 Step 4: Divide both sides by -2. Remember to reverse the inequality sign when dividing by a negative number. (-2x)/(-2) < (-28)/(-2) x < 14 The solution is x < 14. d) 5(2y+9) 3(2y+9) Step 1: Expand both sides of the inequality. 10y + 45 6y + 27 Step 2: Subtract 6y from both sides. 10y - 6y + 45 6y - 6y + 27 4y + 45 27 Step 3: Subtract 45 from both sides. 4y + 45 - 45 27 - 45 4y -18 Step 4: Divide both sides by 4. (4y)/(4) (-18)/(4) y -(9)/(2) y -4.5 The solution is y -4.5. 3. Work out the x and y intercept for the line whose equations are a) y = 4x + 3 To find the x-intercept, set y=0: 0 = 4x + 3 -3 = 4x x = -(3)/(4) To find the y-intercept, set x=0: y = 4(0) + 3 y = 3 The x-intercept is (-(3)/(4), 0) and the y-intercept is (0, 3). b) 3x + 2y = 12 To find the x-intercept, set y=0: 3x + 2(0) = 12 3x = 12 x = (12)/(3) x = 4 To find the y-intercept, set x=0: 3(0) + 2y = 12 2y = 12 y = (12)/(2) y = 6 The x-intercept is (4, 0) and the y-intercept is (0, 6). c) y = -2x + 3 To find the x-intercept, set y=0: 0 = -2x + 3 2x = 3 x = (3)/(2) To find the y-intercept, set x=0: y = -2(0) + 3 y = 3 The x-intercept is ((3)/(2), 0) and the y-intercept is (0, 3). 4. Find the volume and surface of the following a) sphere radius 3.5 cm Volume of a sphere: V = (4)/(3) r^3 Surface area of a sphere: SA = 4 r^2 Given r = 3.5 cm. Step 1: Calculate the volume. V = (4)/(3) × × (3.5 cm)^3 V = (4)/(3) × × 42.875 cm^3 V ≈ 179.59 cm^3 Step 2: Calculate the surface area. SA = 4 × × (3.5 cm)^2 SA = 4 × × 12.25 cm^2 SA ≈ 153.94 cm^2 The volume is 179.59 cm^3 and the surface area is 153.94 cm^2. b) cone radius 7 cm and slant height 10 cm Volume of a cone: V = (1)/(3) r^2 h Surface area of a cone: SA = r (r+l) Given r = 7 cm and l = 10 cm. First, find the height h using the Pythagorean theorem: h = sqrt(l^2 - r^2). h = sqrt((10 cm))^2 - (7 cm)^2 h = sqrt(100 cm)^2 - 49 cm^2 h = sqrt(51) cm ≈ 7.14 cm Step 1: Calculate the volume. V = (1)/(3) × × (7 cm)^2 × sqrt(51) cm V = (1)/(3) × × 49 cm^2 × 7.1414 cm V ≈ 366.49 cm^3 Step 2: Calculate the surface area. SA = × 7 cm × (7 cm + 10 cm) SA = × 7 cm × 17 cm SA = 119 cm^2 SA ≈ 373.85 cm^2 The volume is 366.49 cm^3 and the surface area is 373.85 cm^2. c) cuboid of 5 m x 2 m x 3.2 m Volume of a cuboid: V = lwh Surface area of a cuboid: SA = 2(lw + lh + wh) Given l = 5 m, w = 2 m, h = 3.2 m. Step 1: Calculate the volume. V = 5 m × 2 m × 3.2 m V = 32 m^3 Step 2: Calculate the surface area. SA = 2((5 m × 2 m) + (5 m × 3.2 m) + (2 m × 3.2 m)) SA = 2(10 m^2 + 16 m^2 + 6.4 m^2) SA = 2(32.4 m^2) SA = 64.8 m^2 The volume is 32 m^3 and the surface area is 64.8 m^2. d) cube of sides 4 m Volume of a cube: V = s^3 Surface area of a cube: SA = 6s^2 Given s = 4 m. Step 1: Calculate the volume. V = (4 m)^3 V = 64 m^3 Step 2: Calculate the surface area. SA = 6 × (4 m)^2 SA = 6 × 16 m^2 SA = 96 m^2 The volume is 64 m^3 and the surface area is 96 m^2. 5. A car is brought to rest from 90 km/h in 20 seconds. What is its acceleration? Given: Initial velocity, v_i = 90 km/h Final velocity, v_f = 0 km/h (brought to rest) Time, t = 20 s Step 1: Convert initial velocity from km/

Simplify the following expressions and solve the inequalities: a) y2 y3, b) 50 6, c) (F6 F3) / (F4 F2), d) n7 / n2, e) 56 / 54, f) 9c5 / 9c2, and g) 1/4x - 4 <= 6.

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Simplify the following expressions and solve the inequalities: a) y2 y3, b) 50 6, c) (F6 F3) / (F4 F2), d) n7 / n2, e) 56 / 54, f) 9c5 / 9c2, and g) 1/4x - 4 <= 6.

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