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Given that a = -2, b = -3, c = 2 and d = 0, determine the value of (3abc)^d, a - 5b2 + 4ac, 36ab - 31bc + 2cd, and (3)/(4)a2 + (1)/(2)bc - (1)/(3)d.

Mathematics

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2.2 Given that a=−2a = -2, b=−3b = -3, c=2c = 2 and d=0d = 0, determine the value of the following expression:

2.2.1 (3abc)d(3abc)^d

  1. Substitute the given values into the expression.

    (3×(−2)×(−3)×2)0(3 \times (-2) \times (-3) \times 2)^0

  2. Calculate the value inside the parenthesis.

    (3×6×2)0(3 \times 6 \times 2)^0 (18×2)0(18 \times 2)^0 (36)0(36)^0

  3. Apply the exponent rule x0=1x^0 = 1 for any non-zero xx.

    =1= 1 1\boxed{\text{1}}

    2.2.2 a−5b2+4aca - 5b^2 + 4ac

  4. Substitute the given values into the expression.

    −2−5(−3)2+4(−2)(2)-2 - 5(-3)^2 + 4(-2)(2)

  5. Perform the exponentiation.

    −2−5(9)+4(−2)(2)-2 - 5(9) + 4(-2)(2)

  6. Perform the multiplications.

    −2−45+(−16)-2 - 45 + (-16) −2−45−16-2 - 45 - 16

  7. Perform the subtractions.

    −47−16-47 - 16 =−63= -63 -63\boxed{\text{-63}}

    2.2.3 36ab−31bc+2cd36ab - 31bc + 2cd

  8. Substitute the given values into the expression.

    36(−2)(−3)−31(−3)(2)+2(2)(0)36(-2)(-3) - 31(-3)(2) + 2(2)(0)

  9. Perform the multiplications in each term.

    36(6)−31(−6)+036(6) - 31(-6) + 0 216−(−186)+0216 - (-186) + 0

  10. Perform the addition.

    216+186216 + 186 =402= 402 402\boxed{\text{402}}

    2.2.4 34a2+12bc−13d\frac{3}{4}a^2 + \frac{1}{2}bc - \frac{1}{3}d

  11. Substitute the given values into the expression.

    34(−2)2+12(−3)(2)−13(0)\frac{3}{4}(-2)^2 + \frac{1}{2}(-3)(2) - \frac{1}{3}(0)

  12. Perform the exponentiation and multiplications.

    34(4)+12(−6)−0\frac{3}{4}(4) + \frac{1}{2}(-6) - 0

  13. Perform the remaining multiplications.

    3+(−3)−03 + (-3) - 0

  14. Perform the subtraction.

    3−3−03 - 3 - 0 =0= 0 0\boxed{\text{0}}
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