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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42. Step 1: Rewrite all numbers as powers of their prime factors. 7^2 × (7^2)^-2 × (2 × 7)2^3 × (2^2)^-1 × 7^-2 Step 2: Apply exponent rules (a^m)^n = a^mn and a^m × a^n = a^m+n. 7^2 × 7^-4 × 2^1 × 7^12^3 × 2^-2 × 7^-2 2^1 × 7^2-4+12^3-2 × 7^-2 2^1 × 7^-12^1 × 7^-2 Step 3: Apply exponent rule (a^m)/(a^n) = a^m-n. 2^1-1 × 7^-1 - (-2) 2^0 × 7^-1+2 1 × 7^1 7 The simplified value is 7. 43. Step 1: Convert decimals to fractions or powers of 10. (0.024 × 0.00057)/(0.95 × 0.0018 × 16) = (24)/(1000) × (57)/(100000)(95)/(100) × (18)/(10000) × 16 = (24 × 57 × 100 × 10000)/(1000 × 100000 × 95 × 18 × 16) = (24 × 57)/(10 × 10 × 95 × 18 × 16) Step 2: Simplify the numerical part. = (24 × 57)/(100 × 95 × 18 × 16) Divide 24 by 16: 24/16 = 3/2. = (3 × 57)/(100 × 95 × 18 × 2) Divide 57 by 18: 57/18 = (3 × 19)/(3 × 6) = 19/6. = (3 × 19)/(100 × 95 × 6 × 2) Divide 3 by 6: 3/6 = 1/2. = (19)/(100 × 95 × 2 × 2) Divide 19 by 95: 19/95 = 1/5. = (1)/(100 × 5 × 2 × 2) = (1)/(100 × 20) = (1)/(2000) The value is (1)/(2000). 44. Step 1: A regular heptagon has 7 equal sides and 7 equal interior angles. It can be divided into 7 congruent isosceles triangles, with their vertices at the center O. Step 2: Calculate the angle at the center for each triangle. = (360^)/(7) Step 3: The area of one such triangle (e.g., OAB) is given by the formula: Area of OAB = (1)/(2) r^2 () Given radius r = OB = 10 cm. Area of OAB = (1)/(2) (10 cm)^2 ((360^)/(7)) Area of OAB = (1)/(2) × 100 cm^2 × (51.42857^) Area of OAB = 50 cm^2 × 0.78183 Area of OAB ≈ 39.0915 cm^2 Step 4: Calculate the total area of the heptagon. Area of heptagon = 7 × Area of OAB Area of heptagon = 7 × 39.0915 cm^2 Area of heptagon ≈ 273.6405 cm^2 Step 5: Round to 2 decimal places. Area of heptagon ≈ 273.64 cm^2 The area of the heptagon is 273.64 cm^2. 45. a) Find the number of sides of the polygon; Step 1: Use the formula for the sum of interior angles of a polygon with n sides: Sum of interior angles = (n-2) × 180^ Step 2: Set the formula equal to the given sum and solve for n. (n-2) × 180^ = 1440^ n-2 = (1440)/(180) n-2 = 8 n = 8 + 2 n = 10 The number of sides of the polygon is 10. b) the size of each exterior angle of the polygon. Step 1: The sum of the exterior angles of any convex polygon is 360^. Step 2: For a regular polygon with n sides, each exterior angle is equal. Each exterior angle = (360^)/(n) Step 3: Substitute n=10. Each exterior angle = (360^)/(10) = 36^ The size of each exterior angle is 36^. c) the number of triangles the polygon can be divided into Step 1: A polygon with n sides can be divided into (n-2) triangles by drawing diagonals from one vertex. Step 2: Substitute n=10. Number of triangles = 10 - 2 = 8 The number of triangles the polygon can be divided into is 8. Got more? Send 'em.