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When dealing with expressions raised to a power, it can be very helpful to recognize when they share a common base. In our example, we have the equation \(16^x = 8\). To simplify, both numbers should be expressed as powers of the same base.

Here's how you can do it for this problem:

• Notice that both 16 and 8 can be expressed as powers of 2.
• 16 is the same as \(2^4\) because multiplying 2 by itself four times gives 16.
• Similarly, 8 equals \(2^3\) since multiplying 2 together three times results in 8.
• By expressing both sides of the equation with a common base, things become simpler to solve.

This method of finding a common base makes it straightforward later to compare the exponents directly without any complications in further calculations.

Once we've expressed both sides of our equation using a common base, we can use the power of a power rule to simplify further. The rule states that \((a^m)^n = a^{m \cdot n}\). This is essential when working with exponents that themselves have an exponent.

For our rewritten equation \((2^4)^x = 2^3\), apply the power of a power rule as follows:

• The expression \((2^4)^x\) can be simplified to \(2^{4x}\).
• This operation clears out the parentheses by multiplying the exponents.

Using this rule reduces all the nested expressions to a single power, making the equation much easier to manage.

At this stage, your expression should look somewhat like this: \(2^{4x} = 2^3\). Now we've successfully expressed both sides in terms of a common base (2) and simplified the expressions using the power of a power rule.

Because the bases are the same, you can safely equate the exponents. This means:

• If \(a^m = a^n\), then the relationship \(m = n\) must hold true.

In our problem, this tells us that \(4x = 3\).

Equating exponents is a logical leap that is only valid when both sides of the equation have the same base. Once the equation is reduced to comparing the exponents, it's just a matter of simple algebra to solve, as we isolate \(x\) to find that \(x = \frac{3}{4}\). This step is often the most straightforward in these kinds of problems, once you've set up the equation correctly.