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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Here are the solutions to the problems. Problem 1: Step 1: Convert all mixed numbers to improper fractions. 9(1)/(2) = (19)/(2) 3(1)/(3) = (10)/(3) 6(1)/(4) = (25)/(4) 1(1)/(2) = (3)/(2) Step 2: Evaluate the numerator. The numerator is 9(1)/(2) - 3(1)/(3) ÷ (5)/(9). Substitute the improper fractions: (19)/(2) - (10)/(3) ÷ (5)/(9) Perform the division first: (10)/(3) ÷ (5)/(9) = (10)/(3) × (9)/(5) = (10 × 9)/(3 × 5) = (90)/(15) = 6 Now, perform the subtraction: (19)/(2) - 6 = (19)/(2) - (12)/(2) = (19 - 12)/(2) = (7)/(2) Step 3: Evaluate the denominator. The denominator is (3)/(5) of 6(1)/(4) + 1(1)/(2). "of" means multiplication. Substitute the improper fractions: (3)/(5) × (25)/(4) + (3)/(2) Perform the multiplication first: (3)/(5) × (25)/(4) = (3 × 25)/(5 × 4) = (75)/(20) = (15)/(4) Now, perform the addition: (15)/(4) + (3)/(2) = (15)/(4) + (6)/(4) = (15 + 6)/(4) = (21)/(4) Step 4: Divide the numerator by the denominator. (7)/(2)(21)/(4) = (7)/(2) × (4)/(21) = (7 × 4)/(2 × 21) = (28)/(42) Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 14: (28 ÷ 14)/(42 ÷ 14) = (2)/(3) The evaluated expression is (2)/(3). Problem 2: Step 1: Express all terms in the equation with a base of 2. The equation is 32^(x-3) × 8^(x+4) = 64 ÷ 2^x. 32 = 2^5 8 = 2^3 64 = 2^6 Substitute these into the equation: (2^5)^(x-3) × (2^3)^(x+4) = 2^6 ÷ 2^x Step 2: Apply the exponent rules (a^m)^n = a^mn, a^m × a^n = a^m+n, and a^m ÷ a^n = a^m-n. 2^5(x-3) × 2^3(x+4) = 2^6-x 2^5x-15 × 2^3x+12 = 2^6-x 2^(5x-15) + (3x+12) = 2^6-x 2^8x-3 = 2^6-x Step 3: Equate the exponents since the bases are the same. 8x - 3 = 6 - x Step 4: Solve for x. Add x to both sides: 8x + x - 3 = 6 9x - 3 = 6 Add 3 to both sides: 9x = 6 + 3 9x = 9 Divide by 9: x = (9)/(9) x = 1 The value of x is 1. Problem 3: Step 1: Set the recurring decimal equal to x. Let x = 17.804, which means x = 17.80444... Step 2: Multiply x by a power of 10 to move the non-repeating part to the left of the decimal point. The non-repeating part is '80'. There are two digits. 100x = 1780.444... (Equation 1) Step 3: Multiply x by a power of 10 to move one full repeating block to the left of the decimal point, along with the non-repeating part. The repeating part is '4'. There is one repeating digit. So, we multiply by 10^3 = 1000. 1000x = 17804.444... (Equation 2) Step 4: Subtract Equation 1 from Equation 2. 1000x - 100x = 17804.444... - 1780.444... 900x = 16024 Step 5: Solve for x and simplify the fraction. x = (16024)/(900) Divide both the numerator and the denominator by their greatest common divisor. Both are divisible by 4. x = (16024 ÷ 4)/(900 ÷ 4) = (4006)/(225) Step 6: Convert the improper fraction to a mixed fraction. Divide 4006 by 225: 4006 ÷ 225 = 17 with a remainder 17 × 225 = 3825 Remainder = 4006 - 3825 = 181 So, the mixed fraction is 17(181)/(225). The fraction (181)/(225) is in its simplest form because 181 is a prime number and 225 is not divisible by 181. The recurring decimal as a mixed fraction is 17(181)/(225). 3 done, 2 left today. You're making progress.