Powers of powers and exponential equations
Answers are for parents and teachers.
1
Simplify (4a²b⁻¹)³ ÷ (2a³)² for a ≠ 0, b ≠ 0.
(4a²b⁻¹)³ = 64a⁶b⁻³ and (2a³)² = 4a⁶. Dividing gives 16b⁻³ = 16 ÷ b³.
2
Which is larger, 2⁹ or 3⁶? Use the common exponent 3.
2⁹ = (2³)³ = 8³ = 512 and 3⁶ = (3²)³ = 9³ = 729. So 3⁶ is larger.
3
A message is sent to 3 friends. Each of them sends it to 3 new friends, and so on. In round n, 3ⁿ new people receive it. In which round do 243 people receive it?
3ⁿ = 243 = 3⁵, so n = 5. Check: 3⁵ = 243. It is round 5.
4
Solve 27^(x−1) = 9^(x+1).
3^(3x−3) = 3^(2x+2), so 3x − 3 = 2x + 2 and x = 5. Check: 27⁴ = 3¹² = 9⁶.
5
Solve (3/5)ˣ = 125/27.
125/27 = (5/3)³ = (3/5)⁻³, so x = −3. Check: (3/5)⁻³ = (5/3)³ = 125/27.