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: 1. Evaluate 3x^2 + 4x - 5 given that x=1. Step 1: Substitute x=1 into the expression. 3(1)^2 + 4(1) - 5 Step 2: Perform the calculations. 3(1) + 4 - 5 3 + 4 - 5 7 - 5 = 2 The value of the expression is 2. 2. Evaluate (1)/(x) + (x)/(2x+1) given that x=2. Step 1: Substitute x=2 into the expression. (1)/(2) + (2)/(2(2)+1) Step 2: Simplify the second fraction. (1)/(2) + (2)/(4+1) (1)/(2) + (2)/(5) Step 3: Find a common denominator and add the fractions. The common denominator for 2 and 5 is 10. (1 × 5)/(2 × 5) + (2 × 2)/(5 × 2) (5)/(10) + (4)/(10) (5+4)/(10) = (9)/(10) The value of the expression is (9)/(10). 3. Evaluate (x+y)/(2x+y) + (x)/(y) given that x=1 and y=2. Step 1: Substitute x=1 and y=2 into the expression. (1+2)/(2(1)+2) + (1)/(2) Step 2: Simplify the first fraction. (3)/(2+2) + (1)/(2) (3)/(4) + (1)/(2) Step 3: Find a common denominator and add the fractions. The common denominator for 4 and 2 is 4. (3)/(4) + (1 × 2)/(2 × 2) (3)/(4) + (2)/(4) (3+2)/(4) = (5)/(4) The value of the expression is (5)/(4). 4. Write (10ab)/(9) × (27)/(5abc) in its simplified form and substitute the value to evaluate, given that c=2. Step 1: Simplify the expression by canceling common factors. (10ab)/(9) × (27)/(5abc) = (10 × 27 × ab)/(9 × 5 × abc) Cancel numerical factors: 10/5 = 2, 27/9 = 3. Cancel variable factors: ab/ab = 1. (2 × 3)/(c) = (6)/(c) Step 2: Substitute c=2 into the simplified expression. (6)/(2) = 3 The simplified value is 3. 5. Write (3ab^2)/(8xy) × (16xy^3)/(21ab^2) in its simplified form and substitute the values to evaluate, given that b=1 and y=2. Step 1: Simplify the expression by canceling common factors. (3ab^2)/(8xy) × (16xy^3)/(21ab^2) = (3 × 16 × ab^2 × xy^3)/(8 × 21 × xy × ab^2) Cancel numerical factors: 3/21 = 1/7, 16/8 = 2. Cancel variable factors: ab^2/ab^2 = 1, xy/xy = 1, y^3/y = y^2. (1 × 2 × y^2)/(7 × 1) = (2y^2)/(7) Step 2: Substitute b=1 and y=2 into the simplified expression. Note that b is no longer in the simplified expression. (2(2)^2)/(7) = (2 × 4)/(7) = (8)/(7) The simplified value is (8)/(7). 6. Write (4ax+8x)/(3x-1) + (a+2)/(6x-2) in its simplified form and substitute the value to evaluate, given that x=-1 and a=2. Step 1: Factor the numerators and denominators to simplify. (4x(a+2))/(3x-1) + (a+2)/(2(3x-1)) Step 2: Find a common denominator, which is 2(3x-1). (4x(a+2) × 2)/(2(3x-1)) + (a+2)/(2(3x-1)) (8x(a+2) + (a+2))/(2(3x-1)) Step 3: Factor out (a+2) from the numerator. ((a+2)(8x+1))/(2(3x-1)) Step 4: Substitute x=-1 and a=2 into the simplified expression. ((2+2)(8(-1)+1))/(2(3(-1)-1)) ((4)(-8+1))/(2(-3-1)) ((4)(-7))/(2(-4)) (-28)/(-8) Step 5: Simplify the fraction. (28)/(8) = (7)/(2) The simplified value is (7)/(2). 7. Write (36m^2n^2)/(8m) + (n^3m^2)/(4mn^2) in its simplest form and substitute the values to evaluate, given that m=3 and n=(1)/(3). Step 1: Simplify the first term. (36m^2n^2)/(8m) = (36)/(8) × (m^2)/(m) × n^2 = (9)/(2)mn^2 Step 2: Simplify the second term. (n^3m^2)/(4mn^2) = (1)/(4) × (m^2)/(m) × (n^3)/(n^2) = (1)/(4)mn Step 3: Add the simplified terms. (9)/(2)mn^2 + (1)/(4)mn Step 4: Substitute m=3 and n=(1)/(3) into the expression. (9)/(2)(3)((1)/(3))^2 + (1)/(4)(3)((1)/(3)) (9)/(2)(3)((1)/(9)) + (1)/(4)(3)((1)/(3)) (27)/(18) + (3)/(12) Step 5: Simplify the fractions. (3)/(2) + (1)/(4) Step 6: Find a common denominator and add. (3 × 2)/(2 × 2) + (1)/(4) = (6)/(4) + (1)/(4) = (7)/(4) The simplified value is (7)/(4). 8. Write (3x^2+6x)/(2xy+4y) × (2y)/(9x^2) in its simplified form and substitute the value to evaluate, given that x=1 and y=2. Step 1: Factor the numerators and denominators. (3x(x+2))/(2y(x+2)) × (2y)/(9x^2) Step 2: Cancel common factors. Cancel (x+2) from numerator and denominator. Cancel 2y from numerator and denominator. Cancel 3x from 3x in the numerator and 9x^2 in the denominator (3x/9x^2 = 1/3x). (1)/(1) × (1)/(3x) = (1)/(3x) Step 3: Substitute x=1 and y=2 into the simplified expression. Note that y is no longer in the simplified expression. (1)/(3(1)) = (1)/(3) The simplified value is (1)/(3). That's 2 down. 3 left today — send the next one.