Rationals Between

What fractions can you find between the square roots of 56 and 58?

Root to Poly

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.

Consecutive Squares

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

Power Countdown

Why do this problem?

This problem provides a context for playing with integer and fractional powers so students can get a feel for manipulating indices and become more familiar with commonly used powers of smaller numbers.

Possible approach

Start by showing the first set of numbers and the target of $8$, and set the challenge of making it using only powers and roots. Clarify that a number can be used as a power or a root, so $3$ can be used to cube or cube root a number. There is more than one way of making $8$, so students could be challenged to find as many different ways as possible, or to look for a way that uses all the numbers.

Now give students a target and ask them to choose the numbers they would use to make it, for example $343$. When $7$ and $3$ are suggested, ask if anyone could do it if these numbers were banned...

Once everyone is confident with the concept of using powers and roots to make the target, make a start on the second challenge to make $125$. Students could work in pairs, discussing how to write the target and each of the starting numbers in terms of powers and roots, and writing solutions in terms of fractional powers. In answering the question "How many ways are there?" there is the opportunity for discussion about the order of operations, and why cube rooting followed by squaring is the same as squaring followed by cube rooting, for example.

The last part of the problem is about recognising commonly used powers. The key to finding an answer to those targets which are possible is to spot how to write the target in terms of powers. Students could work on one particular target and then feed back to the rest of the class a solution, or an explanation of how they know their target was impossible.

They could follow this up by making targets of their own and challenging each other to find a way of getting the target.

Key questions

Why is raising to a fractional power the same as finding a root?
Can you spot any numbers which can be written as another number raised to a power?
What makes a total impossible?

Possible extension

Students could make up sets of numbers and targets for each other, using larger powers, and write answers using fractional powers with numerators other than $1$.

Possible support

Give students a selection of numbers and targets to work with which are less than $100$, concentrating mainly on powers of $2$ and $3$.