Step by step
The question
Question: solve 3 to the power (x − 1) = 81.
x is up in the power.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Write 81 as a power of 3
We want both sides to be 3 to a power.
Is 81 made of 3's?
81 is 3 times 3 times 3 times 3: 4 3's multiplied.
So 81 is 3⁴.
Now both sides are 3 to a power.
2 The powers are equal
Both sides are 3 to a power.
3 is the base: the number under the power.
Same base, and the two sides are equal.
So the powers must be equal too.
Each power gives a different number: 9, 27, 81.
So a different power would give a different number.
Now we have x − 1 = 4.
Is x alone now?
Not yet: − 1 is still with x.
How do we remove it?
3 Remove the − 1
We want x by itself.
The − 1 is in the way.
To remove − 1, we add 1.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
− 1 and + 1 cancel each other.
They make zero.
4 + 1 = 5.
So x = 5!
4 Check
Let's check: put 5 in place of x.
We get 81, the same as the right side.
So x = 5 is right!
5 Pick the choice
Our answer is x = 5.
That is choice C.
6 Why not the others
A is 3.
Wrong: 81 = 3⁴, then took 1 away: x − 1 = 4 means we add 1.
B is 4.
Wrong: that is x − 1, not x.
D is 28.
Wrong: 81 ÷ 3 + 1: 81 is 3 to the power 4, not 3 × 27.
Let's look back.
First: write 81 as a power of 3.
Then: the powers are equal.
Then: remove the − 1.
Then: check.
Then: pick the choice.
Last: why not the others.