You’re staring at a calculator or a homework page, and you’re pretty sure you know the answer. Negative times a negative equals a positive, right? So, -5 squared has to be 25. Simple. Except, half the time, your calculator spits out -25 and makes you feel like you’ve lost your mind.
The truth is, both answers can be "correct" depending on how you write the problem. It’s not a math error. It’s a language error. Mathematics has a very specific grammar called the Order of Operations, and most people accidentally ignore the most important rule when dealing with negative numbers.
Honestly, this is the single most common mistake students make in Algebra 1, and it follows them all the way into engineering degrees. If you don't understand the "invisible" parts of the expression, you're going to get the wrong result every single time you use a spreadsheet or a scientific calculator.
Why -5 Squared Isn't Always What You Think
To understand what is -5 squared, we have to look at the hierarchy of operations, often known by the acronym PEMDAS or BODMAS.
In the mathematical world, exponents (that little 2) rank higher than subtraction or negation. When you type $-5^2$ into a high-end TI-84 or even Google’s search bar, the computer sees two distinct parts: the negative sign (negation) and the 5 squared (exponentiation).
Because the exponent comes first, the math looks like this:
$$-(5 \times 5) = -25$$
The negative sign is essentially waiting its turn. It sits there, watching the 5 get multiplied by itself, and then it latches onto the result at the very end. It’s basically saying "the opposite of 5 squared."
The Power of Parentheses
Now, if you want the answer to be 25, you have to change the grammar. You have to use parentheses: $(-5)^2$.
By wrapping the -5 in a "hug," you’re telling the math world that the negative sign and the 5 are a single package deal. They are inseparable. In this case, you are squaring the entire entity of "negative five."
$$(-5) \times (-5) = 25$$
This isn't just a pedantic rule made up by grumpy math teachers. It’s a fundamental part of how programming languages like Python, C++, and Java interpret data. If you’re writing code for a flight stabilizer or a banking app and you miss those parentheses, the logic fails. You’ve got a sign error.
The Calculator Confusion
Not all calculators are created equal. This is where things get messy for the average person.
If you use a basic "four-function" calculator—the kind you find in a junk drawer—and you type 5, then +/-, then square, it might give you 25. Why? Because it’s performing the operations sequentially as you hit the buttons. It’s not looking at the whole expression at once.
However, scientific and graphing calculators are "smart." They use the standard Order of Operations. If you type -5^2, they will strictly follow the rule that Exponents come before Negation (which is treated as a form of multiplication by -1).
- Google Search: Typeset
-5^2and you get -25. - Excel: Type
=-5^2and, interestingly, Excel actually might give you 25.
Wait, what?
Yeah, Microsoft Excel is a bit of an outlier. Historically, Excel’s formula engine treats negation as having a higher priority than exponentiation in certain contexts. This has driven mathematicians crazy for decades. It’s a classic example of "Software Logic" vs. "Mathematical Logic." If you’re moving between a spreadsheet and a physics lab, you have to be incredibly careful about this discrepancy.
Real-World Consequences of a Sign Error
You might think, "Who cares? It’s just a minus sign."
Tell that to the engineers behind the Mars Climate Orbiter. While that specific failure was a metric-to-imperial conversion error, sign errors in squared terms are just as lethal in the world of physics.
Consider the calculation of Kinetic Energy. The formula is:
$$KE = \frac{1}{2} m v^2$$
If $v$ (velocity) is negative because an object is moving in the opposite direction, squaring that velocity must result in a positive value. Energy can't be negative in this context. If a programmer writes a script that calculates $-v^2$ instead of $(v)^2$, the system might think the object is sucking energy out of the environment instead of expending it.
The math breaks. The simulation crashes.
How to Never Get This Wrong Again
The easiest way to think about what is -5 squared is to imagine there is a hidden "1" in front of the minus sign.
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When you see $-5^2$, read it as:
$$-1 \times 5^2$$
You wouldn't multiply by -1 before squaring the 5, right? You’d square the 5 first to get 25, then multiply by -1 to get -25.
If you want the positive version, you must see the negative as part of the number itself, which requires those parentheses. It's a mental shift. You aren't just "squaring a number"; you're following a set of instructions.
The Algebra Perspective
If you’re helping a kid with homework, look at the variables.
Suppose $x = -5$. What is $x^2$?
In this case, the answer is 25. This is because $x$ is -5. When you replace $x$ with its value, you are replacing the whole variable, which implies parentheses: $(-5)^2$.
But if the problem asks for $-x^2$ where $x = 5$, then the answer is -25.
It’s all about the "scope" of the exponent. The exponent is a tiny, powerful king, but it only rules over what is immediately to its left. If there’s a parenthesis, it rules everything inside. If there’s just a number, it only rules that number. It has no power over the negative sign sitting outside its gate.
Practical Steps for Accuracy
If you're working on anything more important than a casual brain teaser, you've got to be disciplined.
- Use parentheses religiously. Even if you think you don't need them, $(-5)^2$ is unambiguous. It communicates your intent to everyone (and every computer) clearly.
- Test your tools. Type
-5^2into your favorite calculator or software right now. See what it does. Knowing the "personality" of your tools is part of being a professional. - Check the context. If you're calculating area, distance, or lighting intensity, and you get a negative number as a result of squaring, you’ve made a syntax error. These physical properties are always non-negative.
- Double-check Excel formulas. If you're building a financial model, manually wrap your negative squares in parentheses to ensure the logic carries over if you ever move that data into a Python script or a different database.
The difference between -25 and 25 isn't just a little dash on the page. It's the difference between a bridge holding up or falling down, or a bank account being in the black or in the red. It's a small detail, but in math, the small details are the only things that actually matter.
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Next Steps for Mastery
To truly lock this in, try a quick experiment. Open three different apps: your phone's default calculator, Google Search, and a spreadsheet. Type in -5^2 and see which ones agree. You'll likely find that the results are split. This simple test is the best way to remind yourself that "math grammar" varies across platforms, and the only way to stay safe is to use parentheses every single time you square a negative number.