Solve 2^(3x) = 4^(x+1) Step by Step

Step by step

The question

Question: solve 2 to the power 3x = 4 to the power (x + 1).

x is up in the powers.

2^3x = 4^{x + 1}

1 Write both as powers of 2

4 is 2².

We write both sides as powers of 2.

2 × 2 = 4

Now both sides are 2 to a power.

2^3x = (2²)^{x + 1}

A power of a power: multiply the powers.

(a^m)ⁿ = a^{m × n}

The powers multiply.

2^3x = 2^{2(x + 1)}

2 The powers are equal

Both sides are 2 to a power.

Same base, and they are equal.

So the powers must be equal too.

same base → same power

Now we have 3x = 2(x + 1).

3x = 2(x + 1)

3 Open the brackets

The 2 in front of the bracket multiplies every part inside.

So we write 2 times each part.

3x = 2 × x + 2 × 1

Work out each multiplication.

3x = 2x + 2

4 Get x on one side

There is x on both sides.

We want x on one side only.

So we remove the 2x from the right side.

To remove 2x, we take away 2x.

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.

3x − 2x = 2x + 2 ~~− 2x~~

2x and − 2x make zero.

Cross them out.

3x − 2x = x.

3x − 2x = x

So x = 2!

x = 2

5 Check

Let's check: put 2 in place of x.

Left: 2³⁽²⁾ = 2⁶ = 64

Now the right side.

Right: 4^{2 + 1} = 4³ = 64

Both sides are 64.

So x = 2 is right!

Let's look back.

First: write both as powers of 2.

Then: the powers are equal.

Then: open the brackets.

Then: get x on one side.

Last: check.

Answer