Solving for x in an exponential equation

Hi Tempest, let's solve this interesting exponential equation together. We are looking for the value of x that satisfies x to the power of two x to the sixth minus nine cubed equals zero.

First, let's move the constant nine cubed to the right hand side of the equation.

Next, we should try to rewrite the right side so it has a base and exponent that look more like the left side. Notice that nine is three squared.

Using the power of a power rule, we multiply the exponents two and three, giving us three to the sixth power.

Now we have a bit of a mismatch between the bases and the exponents. Let's raise both sides of the equation to the third power. This will help align the x to the sixth term on the left.

On the left side, we multiply the outer exponent three with the coefficient two in the exponent. On the right, we multiply six times three to get eighteen.

Let's look at the result: x to the power of six x to the sixth equals three to the eighteenth. We can rewrite the left side using the power rule in reverse.

Now, look at the right side. We want to write eighteen as a product that involves the base. Eighteen is three times six.

Applying the power rule again on the right side, three to the power of three times six is the same as three cubed, all raised to the sixth power.