suppose that a ⊙ b = 5a + 2b. complete parts (a)-(c).\na. g = 5, h = 10, k = 5\nb. g = 2, h = 2, k = 2\nc. g…

suppose that a ⊙ b = 5a + 2b. complete parts (a)-(c).\na. g = 5, h = 10, k = 5\nb. g = 2, h = 2, k = 2\nc. g = 1, h = 2, k = 3\nd. g = 0, h = 0, k = 0\ne. there are no such values where g ⊙ (h ⊙ k)=(g ⊙ h) ⊙ k.\nb. find values for g, h, and k such that g ⊙ (h ⊙ k)≠(g ⊙ h) ⊙ k. choose the correct answer below.\na. g = 1, h = 5, k = 10\nb. g = 1, h = 1, k = 1\nc. g = 4, h = 2, k = 40\nd. g = 2, h = 3, k = 20\ne. there are no such values where g ⊙ (h ⊙ k)≠(g ⊙ h) ⊙ k.

suppose that a ⊙ b = 5a + 2b. complete parts (a)-(c).\na. g = 5, h = 10, k = 5\nb. g = 2, h = 2, k = 2\nc. g = 1, h = 2, k = 3\nd. g = 0, h = 0, k = 0\ne. there are no such values where g ⊙ (h ⊙ k)=(g ⊙ h) ⊙ k.\nb. find values for g, h, and k such that g ⊙ (h ⊙ k)≠(g ⊙ h) ⊙ k. choose the correct answer below.\na. g = 1, h = 5, k = 10\nb. g = 1, h = 1, k = 1\nc. g = 4, h = 2, k = 40\nd. g = 2, h = 3, k = 20\ne. there are no such values where g ⊙ (h ⊙ k)≠(g ⊙ h) ⊙ k.

Answer

Explanation:

Step1: Calculate (h\odot k)

Given (a\odot b = 5a + 2b), then (h\odot k=5h + 2k).

Step2: Calculate (g\odot(h\odot k))

Substitute (a = g) and (b=5h + 2k) into (a\odot b), we get (g\odot(h\odot k)=5g+2(5h + 2k)=5g + 10h+4k).

Step3: Calculate (g\odot h)

(g\odot h = 5g+2h).

Step4: Calculate ((g\odot h)\odot k)

Substitute (a = 5g + 2h) and (b = k) into (a\odot b), we get ((g\odot h)\odot k=5(5g + 2h)+2k=25g+10h + 2k).

Part b

Let's check each option:

  • Option A: If (g = 1), (h = 5), (k = 10) (g\odot(h\odot k)): First, (h\odot k=5\times5 + 2\times10=25 + 20=45). Then (g\odot(h\odot k)=5\times1+2\times45=5 + 90 = 95). ((g\odot h)\odot k): First, (g\odot h=5\times1+2\times5=5 + 10 = 15). Then ((g\odot h)\odot k=5\times15+2\times10=75 + 20=95).
  • Option B: If (g = 1), (h = 1), (k = 1) (h\odot k=5\times1+2\times1=7). (g\odot(h\odot k)=5\times1+2\times7=5 + 14 = 19). (g\odot h=5\times1+2\times1=7). ((g\odot h)\odot k=5\times7+2\times1=35 + 2=37). Here (g\odot(h\odot k)\neq(g\odot h)\odot k).
  • Option C: If (g = 4), (h = 2), (k = 40) (h\odot k=5\times2+2\times40=10 + 80 = 90). (g\odot(h\odot k)=5\times4+2\times90=20+180 = 200). (g\odot h=5\times4+2\times2=20 + 4 = 24). ((g\odot h)\odot k=5\times24+2\times40=120 + 80=200).
  • Option D: If (g = 2), (h = 3), (k = 20) (h\odot k=5\times3+2\times20=15 + 40 = 55). (g\odot(h\odot k)=5\times2+2\times55=10+110 = 120). (g\odot h=5\times2+2\times3=10 + 6 = 16). ((g\odot h)\odot k=5\times16+2\times20=80 + 40=120).

Answer:

b. B. (g = 1), (h = 1), (k = 1)