Taking into account that

$$ 27=3^2 $$

We can rewrite the equation as

Therefore,

Taking into account that

We can rewrite the equation as

Therefore,

Taking into account that

We operate the expression using the properties of powers

Therefore, we have the linear equation

Taking into account that

We can rewrite the equation as

This way, we can extract a common factor of 2 ^x:

Therefore,

Taking into account that

We can rewrite the equation as

We have the common base 3 ^x, but because one of them is squared, we write

Substituting, the equation finishes like

In other words, a quadratic equation:

We multiply the full equation by 9:

We solve it:

Therefore,

So, we obtain

The second option is not possible because it is negative. Therefore,

From where we obtain

Taking into account that

We rewrite the equation as

We call t = 3 ^x:

Substituting, the equation finishes like

We resolve the quadratic equation:

So,

So,

The second solution is not possible because it is negative, but the first one is.

Taking into account that

We rewrite the equation as

We call

Substituting, the equation finishes like

We resolve the previous equation:

Therefore

The second option is not possible. So,

Therefore,

Taking into account that

We rewrite the equation as

We call

Substituting, the equation finishes like

We resolve the quadratic equation:

Therefore

Notice that

So, both are powers of 3.

Then the two solutions are

Taking into account that

We can rewrite the equation as

We call

Substituting, the equation finishes like

We resolve the equation:

So we have

But the solutions

are not possible because one is zero and the other is negative.

So, the only one solution is

Taking into account that

We can rewrite the equation as

Therefore,

Taking into account that

We can rewrite the equation as

We call

Substituting, we obtain a quadratic equation

We resolve it:

So

The first solution is not possible because it is zero. Then,

Taking into account that

we rewrite the equation as

We call

Substituting, we obtain a quadratic equation:

We resolve it:

Therefore,

The first solution is not possible because it is zero.

Therefore, the solution is x = 1.

Notice that

So, the equation can be written as

As we have an exponential dividing, we multiply the full equation by it and that way it disappears:

Now we call

Substituting, the equation finishes like

We resolve it:

Therefore,

The second solution is not possible because it is negative.

So, the solution is

Notice that

So we write the equation as

We call

So,

Substituting, we obtain a quartic equation (equation of degree four):

We resolve it:

The first solution is not possible because it is zero.

So,

Note that

So we can write the equation as

As we have an exponential dividing, we multiply the full equation by it and that way it disappears:

We call

Substituting, we have a cubic equation (degree three)

We rewrite it

We apply Ruffini's Rule:

One of the solutions is t = 4.

We calculate the other two:

But these are not possible solutions because they are negative.

Therefore,

We rewrite the equation:

Therefore, the solution is

We rewrite the equation:

We multiply the full equation by the exponential and that way it disappears:

Therefore,

Note that

We rewrite the equation:

We operate:

Therefore,

Note that

So we rewrite the equation as

We call

Then we obtain the expression

Now we define

Notice that

We are going to suppose that

Therefore

So

And this is not possible.

So, let's suppose

Therefore,

One solution would be t = 0 but, like before, is not possible.

The other solution is

But we have supposed that k = -1 and we have to check that it is true:

Because it's true, the solution to the equation is x = 2.

We rewrite the equation as

We use 25 = 5² and the properties of powers (we write the root in the form of a power):

The, we have

Finally, we resolve the equation:

We write the roots in the form of powers

We want for this to be true:

Therefore, we have two solutions:

$$ x = 0,\ x = -2 $$

We have three nested roots (one inside the other).

We write the roots in the form of powers.

The equations ends up like

We resolve the equation and we obtain the solution

We write the roots in the form of powers:

We want for this to be true:

Finally, we resolve the linear equation:

We write the root in the form of powers:

We want this to be true:

Taking into account that x cannot be 0.

We apply Ruffini's Rule:

One solution is x = 2. We calculate the others:

There aren't any real solutions.

Therefore, the only solution to the exponential equation is

$$ x = 2 $$

We write the roots in the form of powers.

The equation ends up like:

Note that -8 = (-2)³

We want this to be true: