Step by step
The question
Question: solve 5 to the power (x − 1) = 125.
x is up in the power.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Write 125 as a power of 5
We want both sides to be 5 to a power.
Is 125 made of 5's?
125 is 5 times 5 times 5: 3 5's multiplied.
So 125 is 5³.
Now both sides are 5 to a power.
2 The powers are equal
Both sides are 5 to a power.
5 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: 25, 125, 625.
So a different power would give a different number.
Now we have x − 1 = 3.
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.
3 + 1 = 4.
So x = 4!
4 Check
Let's check: put 4 in place of x.
We get 125, the same as the right side.
So x = 4 is right!
5 Pick the choice
Our answer is x = 4.
That is choice C.
6 Why not the others
A is 2.
Wrong: 125 = 5 × 5 × 5: three 5s, not two.
B is 3.
Wrong: that is x − 1, not x.
D is 25.
Wrong: 125 ÷ 5: divided instead of finding the power.
Let's look back.
First: write 125 as a power of 5.
Then: the powers are equal.
Then: remove the − 1.
Then: check.
Then: pick the choice.
Last: why not the others.