Step by step
The question
Question: D = 400 × (0.5)ᵗ is the amount of a drug left, in milligrams, t hours after it is taken.
What does 400 mean?
We write the equation big, so we can label it.
1 What the letters mean
D is the amount of a drug left, in milligrams.
t is the number of hours.
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 hours: 0, 1 and 2.
0 is where it starts.
t = 0: no hours have passed, so we multiply by 0.5 zero times.
That is why 0.5 to the power 0 is 1.
400 times 1 is 400.
t = 1: 0.5 to the power 1 is just 0.5.
400 times 0.5 is 200.
t = 2: 0.5 squared is 0.5 times 0.5.
400 times 0.5 is 200.
Times 0.5 again: 200 times 0.5 is 100.
Look at the values of D: 400, 200, 100.
Each time t goes up 1, D 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 hour.
That is the power t: it counts the hours.
400 is D at t = 0: the first number in the table.
So 400 is the amount of a drug left at the start.
Careful!
400 is not added each hour.
It is the amount at t = 0, before any multiplying.
4 Say it in words
So 400 means: the amount of a drug left is 400 milligrams at t = 0.
That is the start.
5 Check
Let's check by going back one hour, from t = 1 to t = 0.
To get to t = 1, we multiplied by 0.5.
To get to t = 1, we took half.
So the start is double 200: 200 times 2 is 400: 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.