Step by step
The question
Question: solve 2 times 5 to the power x, − 1 = 9.
x is up in the power.
Careful!
The power is only on the 5.
Powers come before times.
So we can't join 2 and 5 first.
1 Get the power alone
First the power must be alone: nothing added to it, nothing times it.
We want the power 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 make zero.
Cross them out.
9 + 1 is 10.
Now we have 2 × 5ˣ = 10.
The power is times 2.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just the power is left.
10 divided by 2 is 5.
Now we have 5ˣ = 5.
2 Write 5 as a power of 5
5 is 5 times itself, 1 times.
So 5 is 5¹.
Now both sides are 5 to a power.
3 The powers are equal
Both sides are 5 to a power.
Same base, and they are equal.
So the powers must be equal too.
So x = 1!
4 Check
Let's check in the question: put 1 in place of x.
We get 9, the same as the right side.
So x = 1 is right!
Let's look back.
First: get the power alone.
Then: write 5 as a power of 5.
Then: the powers are equal.
Last: check.