Let's just be honest for a second. Most of us haven't thought about exponents since we were sitting in a stuffy algebra classroom wondering when we’d ever need to know the difference between a base and a power in the "real world." Then you're hit with a question like what is 3 to the third power while helping a kid with homework or trying to scale a recipe, and suddenly, that mental rust is very real.
The answer is 27.
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Simple, right? But the way we get there—and why our brains often try to tell us the answer is 9—is actually pretty fascinating if you look at how humans process patterns. We’re wired for addition. Multiplication is a bit of a stretch. Exponents? That’s where things get weird. When you see 3 to the third power, your brain wants to take a shortcut. It sees two threes and screams "Nine!" but that’s the trap of linear thinking in an exponential world.
The Mechanics of Cubing
When we talk about 3 to the third power, we're dealing with what mathematicians call "cubing." It’s represented as $3^3$. The bottom number is your base. The little floating number is your exponent. Essentially, you aren't multiplying the base by the exponent; you’re using the base as a factor the number of times indicated by the exponent.
So, it's $3 \times 3 \times 3$.
Three times three is nine. Then you take that nine and multiply it by three one more time. Boom. Twenty-seven. It grows much faster than people expect. This is the fundamental building block of volume in geometry. If you have a physical cube that is three inches wide, three inches deep, and three inches tall, you have exactly 27 cubic inches of space. It’s a physical reality, not just a scribbled note on a chalkboard.
Why 27 is a Special Number
In the world of mathematics, 27 is what’s known as a perfect cube. It’s the only positive integer that is three times the sum of its digits ($2 + 7 = 9$, and $9 \times 3 = 27$). If you’re into recreational math or number theory, these kinds of "coincidences" are like finding a four-leaf clover. It’s also a Smith number in base 10, meaning the sum of its digits is equal to the sum of the digits of its prime factors.
Numbers aren't just quantities. They have personalities.
Common Pitfalls and Why We Fail at Mental Math
Most people who miss this question do it because they are rushing. They see the numbers and perform the most basic operation available: $3 \times 3$. In psychology, this is often attributed to cognitive load. If you're stressed or distracted, your brain defaults to the path of least resistance.
Teachers see this constantly. A student knows how to do the work, but under the pressure of a timed test, $3^3$ becomes 9. It’s a classic "System 1" thinking error, as Daniel Kahneman might describe it in Thinking, Fast and Slow. Your fast, intuitive brain makes a guess before your slow, logical brain can step in and say, "Wait, we need to multiply by three again."
The Exponential Curve
To really grasp the power of $3^3$, you have to look at the sequence.
- $3^1 = 3$
- $3^2 = 9$
- $3^3 = 27$
- $3^4 = 81$
- $3^5 = 243$
Notice how quickly the gaps grow? The jump from power one to power two is only 6 units. The jump from power two to power three is 18 units. By the time you get to $3^5$, you’ve cleared two hundred. This is the essence of exponential growth. It’s why compounding interest in a savings account is so powerful over decades, and why a virus can spread through a population so quickly. Small changes at the start lead to massive, world-altering differences down the line.
Real World Applications of Powers of Three
You might think you never use this. You’d be wrong. If you work in computer science or data storage, you're constantly dealing with powers—usually powers of two, but ternary logic (base 3) is a real, albeit niche, field of computing. Some researchers believe ternary computing could be more efficient than the binary systems we use today.
In music theory, the relationship between frequencies often involves powers. A perfect fifth, for example, has a frequency ratio of 3:2. When you start stacking these intervals (the "Circle of Fifths"), you're essentially playing with the math of threes.
Even in something as mundane as a bracket-style tournament, like March Madness, you’re looking at powers. While those are usually powers of two (64 teams, 32, 16...), if you were to design a "Triple Threat" tournament where three teams played at once and only one moved on, you’d be calculating your rounds based on what is 3 to the third power and beyond.
Testing Your Mental Agility
Try this next time you’re bored. Instead of just stopping at 27, try to double it and then triple it. It’s a great way to keep the prefrontal cortex sharp. We spend so much time letting calculators do the heavy lifting that we lose the "feel" for numbers. Understanding that 27 is the result of three threes gives you a better intuitive sense of scale.
If someone says, "It’s increased three-fold, three times," you now know that isn't a 900% increase; it's a 2700% increase from the original base. That’s a massive distinction in business or science.
The Logic of the Base System
We live in a base-10 world. We have ten fingers, so we count to ten. But imagine if we lived in a base-3 world. In a "ternary" system, the number 27 would be written as 1000. It sounds confusing because we aren't used to it, but it's mathematically "cleaner" in certain complex physics simulations.
Thomas Fowler, a self-taught mathematician in the 1800s, actually built a ternary calculating machine because he found it more efficient for certain types of bookkeeping. He was way ahead of his time. He realized that powers of three offer a unique balance that powers of two don't quite hit.
Practical Steps for Mastering Exponents
If you want to never get tripped up by this again, stop thinking about exponents as "math problems" and start thinking about them as "growth stories."
- Visualize the Cube. Literally picture a 3x3 square on the ground. That’s 9 tiles. Now, stack three of those squares on top of each other. You can see the 27 tiles in your mind's eye. This spatial awareness sticks better than a memorized formula.
- Use the "Plus One" Rule. Whenever you see an exponent, remind yourself: "I have to do the multiplication one more time than I think." If it’s $3^3$, you do $3 \times 3$, and then—crucially—add that one extra step.
- Practice the First Five. Memorize the first five powers of 2, 3, and 5. These are the "Big Three" of mental math. Knowing that $3^3 = 27$ and $3^4 = 81$ off the top of your head will make you look like a wizard in meetings.
- Relate to Money. If you invested 3 dollars and it tripled every year, in three years you'd have 27 dollars. It’s a small amount, but it demonstrates the principle of $3^3$ in a way that feels tangible.
Math isn't just about getting the "right" answer for a test. It's about understanding the underlying structure of the universe. Whether you're calculating the volume of a box or understanding the rate of a chemical reaction, the power of three is everywhere. The next time someone asks you what is 3 to the third power, you won't just say "27"—you'll actually understand why it matters.