Negative powers
What does this number mean? Which order of 1, 2, 3 and 4 makes the highest value? Which makes the lowest?
Problem
Negative powers printable worksheet
Take a look at this expression. What does it mean? $$\left(\left(-4^{-3}\right)^{-2}\right)^{-1}$$
Do you think it means
$\left(\left(\left(-4\right)^{-3}\right)^{-2}\right)^{-1}$ (Interpretation A) or $\left(\left(-\left(4^{-3}\right)\right)^{-2}\right)^{-1}$(Interpretation B)?
Check your thinking
By usual mathematical conventions for orders of operations, $\left(\left(-4^{-3}\right)^{-2}\right)^{-1}$ means $\quad\left(\left(-\left(4^{-3}\right)\right)^{-2}\right)^{-1}$ (Interpretation B). You can include the extra brackets if you find these helpful, but make sure you put them in the right place.
Check your thinking
Both $\left(\left(\left(-4\right)^{-3}\right)^{-2}\right)^{-1}$ and $\left(\left(-\left(4^{-3}\right)\right)^{-2}\right)^{-1}$ have the same value, so no, in this case it doesn't make a difference.
Check your thinking
If we swap the order of $3$ and $2$, then Interpretation A gives us
$$\left(\left((-4)^{-2}\right)^{-3}\right)^{-1}=\left(\left(\dfrac{1}{(-4)^{2}}\right)^{-3}\right)^{-1}=\left(\dfrac{1}{16}\right)^{(-3)\times(-1)}=\left(\dfrac{1}{16}\right)^3=\dfrac{1}{16^3}$$
But Interpretation B gives us
$$\left(\left(-(4^{-2})\right)^{-3}\right)^{-1}=\left(\left(-\dfrac{1}{4^{2}}\right)^{-3}\right)^{-1}=\left(-\dfrac{1}{4^{2}}\right)^{(-3)\times(-1)}=\left(-\dfrac{1}{16}\right)^3=-\dfrac{1}{16^3}$$
So in this case, it does matter how we interpret $\left(\left(-4^{-2}\right)^{-3}\right)^{-1}$.
Getting Started
There are lots of ideas about powers that come together in this problem. Here are some ideas to think about as you work your way into the problem.
Getting a sense of powers
What does $4$ to the power $3$ mean?
What does $4$ to the power $2$ mean?
What does $4$ to the power $1$ mean?
What does $4$ to the power $0$ mean?
What does $4$ to the power $-1$ mean?
What does $4$ to the power $-2$ mean?
What does $4$ to the power $-3$ mean?
Getting a sense of powers of negative numbers
What does $-4$ to the power $3$ mean?
What does $-4$ to the power $-3$ mean?
Combining powers
There are lots of ways to combine powers in this problem.
For example, if you tried to work out the value of $\left(\left(-(4^{-3})\right)^{-2}\right)^{-1}$ you could work this out as:
$$\left(\left(-(4^{-3})\right)^{-2}\right)^{-1}=\left(\left(-\dfrac{1}{4^{3}}\right)^{-2}\right)^{-1}=\left(\left(-\dfrac{1}{64}\right)^{-2}\right)^{-1}=\left((-64)^2\right)^{-1}=\left(4096\right)^{-1}=\dfrac{1}{4096}$$
Or you could try something more like this:
$$\left(\left(-(4^{-3})\right)^{-2}\right)^{-1}=\left(\left(-\dfrac{1}{4^{3}}\right)^{-2}\right)^{-1}=\left(-\dfrac{1}{4^{3}}\right)^{(-2)\times(-1)}=\left(-\dfrac{1}{64}\right)^2=\dfrac{1}{64^2}=\dfrac{1}{4096}$$
Why do the rules for indices mean that both these approaches must give the same value?
Could you leave the powers of $4$ and $64$ as $4^3$ and $64^2$ and still see that these methods lead to the same values?
How could you use rules for indices to find more efficient approaches when working with these and similar expressions?
Teachers' Resources
Why do this problem?
This problem is a good way to illustrate the precision of meaning that mathematics often requires - in this case the precedence of the operations.
The task provides some practice at interpreting negative indices and powers of negative numbers, but more deeply it invites students to explore some general ideas about powers and how to simplify the 'stacked' indices.
Possible approach
It would be best to avoid reliance on calculators or spreadsheets when working on this task, but if you think calculators would be helpful you could ask students to think carefully about how their calculators are interpreting the expressions.
This problem has two main stages. The first part focuses on the meaning of the expression $\left(\left(-4^{-3}\right)^{-2}\right)^{-1}$ and draws attention to mathematical conventions. Students may already be familiar with orders of brackets, powers and multiplication, but this is quite a complicated expression for them to make sense of.
Share the initial expression. Ask students to think about how they interpret it and discuss this with a neighbour. Students might discuss the two different expressions given in the problem (interpretations A and B), so you could move the discussion towards these.
Make sure students understand that the mathematical convention is to interpret the starting expression as 4, rather than -4, being raised to the power of -3.
If students haven't already done so, ask them to work out the values of the expressions that come from each interpretation. They will find that they get the same values in this case, but then ask them to try reordering the indices to see if this is always the case. Encourage students to use what they know about powers here and think about what might affect the value, rather than just working through all the possibilities.
Reveal that swapping -2 and -3 gives different values for the two interpretations. You could give students some time to discuss this to make sure they have made sense of why the two interpretations give different values in this case. They might start to make generalisations at this stage, drawing attention to the parity of the powers, and how they affect whether the final solution will be positive or negative.
Now move on to the second stage of the problem, where students can change the order of 1, 2, 3 and 4. Emphasise that students should use the conventional interpretation of the expressions.
The printable version of this problem has the Check your thinking text on the second page, so that you can decide whether to make this available to students if you use the printable version.
Key questions
What order do you need to apply the operations in to work out the value of this expression?
What rules for manipulating indices do you know?
How could you start to simplify these expressions?
What happens if you raise a negative number to an even power? Or an odd power?
Possible support
Students might find it helpful to work through one or two calculations fully, writing out each step, so that they see how the negative numbers and odd or even powers are combined in each case. They might do this in more than one way as they recall different laws of indices.
Possible extension
Students might go on to consider which other sets of four values will produce the same 'stacked' value, irrespective of stacking order.
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