Welcome back to Rules of Integer Exponents! You've already added two essential exponent shortcuts to your toolkit: the product rule for multiplying same-base powers and the quotient rule for dividing them. Now past the halfway mark of this course, we tackle a new kind of expression: what happens when a power is raised to another power? This is the Power of a Power Rule, and together with the product and quotient rules, it completes the core set of exponent laws.

So far, we've combined powers that sit next to each other in a multiplication or a division. But sometimes one power sits inside another. The expression $(2^3)^2$ tells us to take $2^3$ and raise that entire quantity to the second power — we're stacking one exponent on top of another.

This kind of structure shows up whenever a repeated process is itself repeated. Think of folding a piece of paper: each fold doubles the number of layers, so folding three times gives layers. If you then repeated that entire three-fold sequence twice, the layers would compound. Let's unpack what happens by expanding a few examples by hand.

Let's start with $(2^3)^2$. The outer exponent $2$ means we multiply the inner power, $2^3$, by itself two times:

Here is the general rule we just discovered:

When a power is raised to another power, keep the base and multiply the exponents. A few quick examples:

The rule holds whenever either exponent is zero. Because multiplying any number by zero gives zero, and any nonzero base to the zero power equals 1, the result is always 1:

The power of a power rule appears naturally whenever we square or cube a quantity that is already expressed as a power. Let's look at two quick examples.

Area of a square garden. A garden plot measures $5^2 = 25$ feet on each side. To find the area, we square the side length:

Here's the key takeaway: when a power is raised to another power, keep the base and multiply the exponents — $(a^m)^n = a^{\,m \times n}$. This works for positive, zero, and negative exponents alike. Combined with the product rule and the quotient rule from the first two lessons, you now have the complete set of core exponent laws.

Up next, you'll put this rule to the test! The practice exercises will walk you through hands-on expansions, direct exponent multiplication, negative exponent challenges, and real-world scenarios spanning digital imaging, chip engineering, and pharmacy. Let's get to it!