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An operation is defined on any three real numbers by r o+s o+t=r^(s-2t). If 3o+5o+m=81, then m= ?

An operation is defined on any three real numbers by r⊕s⊕t=rs−2t r \oplus s \oplus t=r^{s-2 t} . If 3⊕5⊕m=81 3 \oplus 5 \oplus m=81 , then m= m= ?

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Full solution

Q. An operation is defined on any three real numbers by r⊕s⊕t=rs−2t r \oplus s \oplus t=r^{s-2 t} . If 3⊕5⊕m=81 3 \oplus 5 \oplus m=81 , then m= m= ?
  1. Identify Operation: Identify the operation defined by r∘s+s∘t=r(s−2t)r \circ s + s \circ t = r^{(s-2t)}.
  2. Apply Operation: Apply the operation to 3o+5o+m=813o+5o+m=81, which means 35−2m=813^{5-2m}=81.
  3. Recognize Value: Recognize that 8181 is 343^4.
  4. Set Up Equation: Set up the equation 35−2m=343^{5-2m}=3^4.
  5. Set Exponents Equal: Since the bases are the same, set the exponents equal to each other: 5−2m=45-2m=4.
  6. Solve for m: Solve for m: 5−4=2m5-4=2m.
  7. Final Solution: Divide both sides by 22 to get m=12m=\frac{1}{2}.

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