Step by step
The question
Question: H = 64(0.5)ᵇ is the height, in feet, of a ball after b bounces.
What does 64 mean?
We write the equation big, so we can label it.
1 What the letters mean
H is the height of a ball, in feet.
b is the number of bounces.
It is the small raised number, the power: how many times we multiply by 0.5.
2 Make a small table
To see what each number does, we try a few bounces: 0, 1 and 2.
0 is where it starts.
b = 0: no bounces have passed, so we multiply by 0.5 zero times.
That is why 0.5 to the power 0 is 1.
64 times 1 is 64.
b = 1: 0.5 to the power 1 is just 0.5.
64 times 0.5 is 32.
b = 2: 0.5 squared is 0.5 times 0.5.
64 times 0.5 is 32.
Times 0.5 again: 32 times 0.5 is 16.
Look at the values of H: 64, 32, 16.
Each time b goes up 1, H is multiplied by 0.5.
The same number every time.
3 Find the numbers in the table
Now we write the equation again, and label its numbers.
0.5 is what we multiply by, once for each bounce.
That is the power b: it counts the bounces.
64 is H at b = 0: the first number in the table.
So 64 is the height of a ball at the start.
Careful!
64 is not added each bounce.
It is the amount at b = 0, before any multiplying.
4 Say it in words
So 64 means: the height of a ball is 64 feet at b = 0.
That is the start.
5 Check
Let's check by going back one bounce, from b = 1 to b = 0.
To get to b = 1, we multiplied by 0.5.
To get to b = 1, we took half.
So the start is double 32: 32 times 2 is 64: the start.
It works!
Let's look back.
First: what the letters mean.
Then: make a small table.
Then: find the numbers in the table.
Then: say it in words.
Last: check.