Why use this resource?

This offers a thought-provoking challenge, but one which is accessible if students are given some time to think about the problem. It’s great for practising both quadratics and laws of indices, and students can get a lot from making sure that they find all the solutions (there are more than three solutions for the first part).
There is the potential for a rich discussion based on this resource. This could touch on finding systematic approaches to unfamiliar or seemingly complex equations, independence and constraints in the different cases, as well as what we can or can’t say about \(0^0\).

Possible approach

Start with just the first equation. This will help to establish the three cases that need to be considered in each part of the question. Students should have the opportunity to share ideas, strategies and solutions after working on the first equation because some may have more solutions than others. When starting the second equation, students could be encouraged to think about which approaches from the first equation can be applied in this case.

Key questions

  • If you know that \(a^b=1\), what can you say about \(a\) and \(b\)?

  • What if \(a^b=1\) and \(b=2\)?

Possible extension

For a real challenge (requiring a bit more knowledge), students could consider finding the complex solutions.

A version of this resource has been featured on the NRICH website. You might like to look at some students’ solutions that have been submitted there.