Step by step
The question
Question: solve 2 to the power (x + 3) = 64.
x is up in the power.
There are four choices.
First we solve it, the normal way.
Then we pick.
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 + 3 = 6.
Is x alone now?
Not yet: + 3 is still with x.
How do we remove it?
3 Remove the + 3
We want x by itself.
The + 3 is in the way.
To remove + 3, we take away 3.
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.
+ 3 and − 3 cancel each other.
They make zero.
6 − 3 = 3.
So x = 3!
4 Check
Let's check: put 3 in place of x.
We get 64, the same as the right side.
So x = 3 is right!
5 Pick the choice
Our answer is x = 3.
That is choice A.
6 Why not the others
B is 6.
Wrong: that is x + 3, not x.
C is 9.
Wrong: 6 + 3: added 3 instead of taking it away.
D is 32.
Wrong: 64 ÷ 2: divided instead of finding the power.
Let's look back.
First: write 64 as a power of 2.
Then: the powers are equal.
Then: remove the + 3.
Then: check.
Then: pick the choice.
Last: why not the others.