2 × 2 × 2 × 2 is written 2⁴: the base 2 is multiplied by itself, and the exponent 4 counts how many times. 2⁴ = 16, not 8.
Fold a sheet of paper in half, then again, and again. After 7 folds it is 2⁷ = 128 layers thick. Powers grow fast, and the short way to write them is exponent notation.
When the same number is multiplied again and again, there is a short way to write it:
2×2×2×2=24
The base (2) is the number being multiplied. The exponent (4), also called the index or power, counts how many times it appears. You say "2 to the power of 4". You have met two already: 52 is "5 squared", and 23 is "2 cubed".
The board grows a factor tree: it splits a number into a factor pair (two numbers that multiply to it), and keeps splitting until every branch is a prime, a number with exactly two factors, 1 and itself (2, 3, 5, 7, …).
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
2 cubed
How many 2s are multiplied in 23?
TryTry it on the board and compare the result with your prediction.
Grow a tree for 16
Click a number that is not prime (or use the Split buttons), then choose a factor pair.
Try it yourselfYour edits are saved in this browser.
Split 16 into factor pairs until every branch ends in a prime. How many 2s? Was your prediction for 23 right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Base and exponent
Here is how to go between the two forms.
Worked exampleBefore each step, say what you think comes next.
2×2×2×2×2=25Count the equal factors: five 2s, so the exponent is 5.
25=2×2×2×2×2=32Expand and multiply: 4, 8, 16, 32.
3×3×5=32×5Different bases get their own powers.
104=10000Powers of 10: the exponent counts the zeros.
Now try one: 34.
Read Mia's working. What does the 5 count?
Work out 34.
Mia works out 25. Click the line where her working goes wrong.
Match each power to what it means.
$10^3$
$3^2$
$2^3$
$5^1$
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
Start with 2 × 16, or 4 × 8.
Try it yourselfYour edits are saved in this browser.
Grow a tree for 32. Write 32 as a power of 2.
Use the board. This ticks when you get there.
Work out 23×32.
PractiseA few on your own, to check you have it.
Powers
Work out 43.
Work out 105.
Review: work out 144.
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
64 three ways
Then write 64 as a power of 2, a power of 4 and a power of 8.
Try it yourselfYour edits are saved in this browser.
Need a hint?
Split 64 into 8 × 8, or 2 × 32, or 4 × 16.
Another hint
Group the six 2s in pairs for 4s, in threes for 8s.
Grow a tree for 64.
Use the board. This ticks when you get there.
For teachers
Students grow factor trees for powers of 2 to see repeated equal factors, read and write exponent notation (base and exponent), evaluate powers, and meet powers of 10 (AC9M7N02, node 1). About 40 minutes.
I can write a repeated multiplication as a power, and expand a power.
I can say what the base and the exponent mean.
I can work out powers, including powers of 10.
Before class
Do the Run board yourself first. Students need squares (previous lessons) and primes and factor pairs (Year 6).
The one idea: the exponent counts how many times the base is multiplied. The common mistake is multiplying base and exponent (2⁵ = 10, Mia's mistake). Fluency with the notation is the hinge for prime factorisation in the next lesson.
The factor tree board is used again for prime factors and HCF and LCM.
Running the lesson
Predict: students who chose 6 multiplied 2 × 3. Ask them to write 2³ the long way.
Run: ask how many 2s the tree ended with, and where the 4 in 2⁴ comes from.
Investigate: link 10³ to place value: thousands.
Modify: for 2⁵ against 5², ask for a prediction before working them out.
Make: collect the three names for 64 and ask for a fourth (64¹).
The hard part (on purpose)
Reading the small raised number as a count, not a multiplier. Ask students to write the power the long way first, every time, before working it out.
When students get stuck
A student multiplies base and exponent (Mia's mistake). Ask: Write it the long way first. How many 2s? Fix: Expand, then multiply one factor at a time.
A student swaps base and exponent (7³ written 3⁷). Ask: Which number is being multiplied? Fix: The base is the big number; the small raised number counts.
A student adds different bases (2² × 5³ as 7⁵). Ask: Are the factors the same number? Fix: Only equal factors share a power; each base gets its own.
Stretch and support
Stretch: The fold-the-paper hook: after how many folds is the stack more than 1000 layers? More than 1 000 000?
Support: Expand every power before working it out; use the tree board for powers of 2 and 3.
Where it leads
Next: Prime factors (node 2), writing any number as a product of prime powers.
Where it leads: index laws and scientific notation (Year 8 and 9), and square numbers as powers with exponent 2.
What each step is for
2 cubed: Committing to a value for 2³ shows whether they read the exponent as a multiplier (6) or as a count of factors (8).
grow a tree for 16: Splitting 16 until every branch is a prime shows four equal factors, and the board writes them as 2⁴.
base and exponent: The exponent counts factors of the base; Mia's mistake multiplies the base by the exponent. Powers of 10 count zeros, the same idea in place value.
one change at a time: One change at a time: a bigger tree, writing a power from a product, two bases, then the exponent 1 and the order of base and exponent.
powers: Evaluating, writing and comparing powers, mixed; the exponent as a multiplier is the planted slip; the review keeps roots fresh.
64 three ways: Growing a tree for 64 and writing it as a power of 2, of 4 and of 8 shows one number has several power names, all from the same six 2s.
Now you can
Write a repeated multiplication as a power, and expand a power.