Step by step
The question
Question: solve 2 to the power (x − 1) = 64.
x is up in the power.
1 Write 64 as a power of 2
We want both sides to be 2 to a power.
Is 64 made of 2's?
64 is 2 times 2 times 2 times 2 times 2 times 2: 6 2's multiplied.
So 64 is 2⁶.
Now both sides are 2 to a power.
2 The powers are equal
Both sides are 2 to a power.
2 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: 4, 8, 16.
So a different power would give a different number.
Now we have x − 1 = 6.
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.
6 + 1 = 7.
So x = 7!
4 Check
Let's check: put 7 in place of x.
We get 64, the same as the right side.
So x = 7 is right!
Let's look back.
First: write 64 as a power of 2.
Then: the powers are equal.
Then: remove the − 1.
Last: check.