What Does Squared Mean In Math? Plus Squared Times Squared
The short version
- Squared means a number multiplied by itself. 5 squared is written 52 and equals 5 × 5 = 25.
- It’s called “squared” because a square with sides of 5 has an area of 25.
- Squared is not the same as times 2. 62 = 36, but 6 × 2 = 12.
- Squared times squared adds the exponents: x2 × x2 = x4. But x2 + x2 = 2x2.
- A negative number squared is positive: (-4)2 = 16.
Squared means multiplying a number by itself. When you see a small raised 2, like 72, read it as “7 squared” and compute 7 × 7 = 49. The big number is the base and the small 2 is the exponent, which tells you how many copies of the base to multiply together.
That covers the basics. The rest of this guide handles what homework actually throws at kids: squaring negatives and fractions, why “squared” isn’t “times 2,” and the algebra rules for squared times squared.
What squared means, and why it’s called that
Draw a square that’s 4 units on each side. Its area is 4 × 4 = 16 square units. That’s where the name comes from: 4 squared is the area of a square with side 4.
So when you see a number squared, you can picture a square made of little unit tiles. 32 is a 3 by 3 grid of 9 tiles. 102 is a 10 by 10 grid of 100 tiles.
The numbers you get from squaring whole numbers are called perfect squares. Kids are expected to know the first dozen or so by heart:
| n | n2 | n | n2 | n | n2 |
|---|---|---|---|---|---|
| 1 | 1 | 6 | 36 | 11 | 121 |
| 2 | 4 | 7 | 49 | 12 | 144 |
| 3 | 9 | 8 | 64 | 13 | 169 |
| 4 | 16 | 9 | 81 | 14 | 196 |
| 5 | 25 | 10 | 100 | 15 | 225 |
Squared vs. times 2 vs. square root
| Operation | Written | With 6 |
|---|---|---|
| Squared | 62 | 6 × 6 = 36 |
| Times 2 (doubling) | 6 × 2 | 12 |
| Square root | √36 | 6, because 6 × 6 = 36 |
Square root undoes squaring. If 6 squared is 36, the square root of 36 is 6.
Squaring negatives, fractions and decimals
Negative numbers
A negative times a negative is positive, so squaring any negative number gives a positive answer.
(-4)2 = (-4) × (-4) = 16
Watch the parentheses, though. Without them, the exponent only applies to the 4, and the minus sign waits until after:
-42 = -(4 × 4) = -16
This one catches kids and adults alike. Calculators and spreadsheets don’t all handle it the same way, so type the parentheses whenever you mean to square a negative.
Fractions
Square the top and square the bottom.
(3/4)2 = (3 × 3)/(4 × 4) = 9/16
Decimals
0.52 = 0.5 × 0.5 = 0.25
1.22 = 1.2 × 1.2 = 1.44
Notice that 0.5 squared is smaller than 0.5. Squaring a number between 0 and 1 always makes it smaller, which surprises a lot of kids who think squaring always makes things bigger.
Squared times squared: the exponent rules
Once kids hit pre-algebra, “squared” shows up with variables. Here are the rules, each with a number check so you can see it’s true.
Multiplying: add the exponents
x2 × x2 = x2+2 = x4
Why: x2 is x × x, so x2 × x2 is x × x × x × x, which is four x’s multiplied together.
Check with 3: 32 × 32 = 9 × 9 = 81, and 34 = 3 × 3 × 3 × 3 = 81. They match.
The same rule works for other exponents: x2 × x3 = x5.
With numbers in front: multiply those, add the exponents
2x2 × 3x2 = (2 × 3)(x2 × x2) = 6x4
Squaring a square: multiply the exponents
(x2)2 = x2 × 2 = x4
Check with 3: (32)2 = 92 = 81 = 34.
Adding: the exponent stays the same
x2 + x2 = 2x2
Adding like terms is counting them: one x2 plus another x2 is two of them. Check with 3: 9 + 9 = 18, and 2 × 9 = 18. (And 34 = 81, so x4 would be wrong.)
Squaring a product and a sum
(xy)2 = x2y2. You can square each factor.
(x + 3)2 is different. It means (x + 3)(x + 3), which is x2 + 6x + 9. Check with x = 2: (2 + 3)2 = 25, and 4 + 12 + 9 = 25. The tempting answer, x2 + 9, gives 13. Wrong.
| Expression | Simplifies to | Rule |
|---|---|---|
| x2 × x2 | x4 | Add exponents |
| (x2)2 | x4 | Multiply exponents |
| x2 + x2 | 2x2 | Combine like terms |
| (xy)2 | x2y2 | Square each factor |
| (x + 3)2 | x2 + 6x + 9 | Multiply it out |
Where squared shows up
- Area. A square room 12 feet on a side is 122 = 144 square feet. The “square” in square feet is the same idea.
- The Pythagorean theorem. In a right triangle, a2 + b2 = c2. With legs 3 and 4: 9 + 16 = 25, so the long side is √25 = 5.
- Graphs. y = x2 makes the U-shaped curve called a parabola, a big part of Algebra 1.
Common mistakes with squared
- Doubling instead of squaring. 82 is 64, not 16.
- Losing the sign. (-5)2 = 25, but -52 = -25.
- Adding exponents when adding terms. x2 + x2 is 2x2, never x4.
- Multiplying exponents when multiplying terms. x2 × x3 is x5, not x6.
- Squaring a sum term by term. (x + 3)2 is not x2 + 9.
Practice problems
- What is 7 squared?
- What is 122?
- Find (-6)2 and -62.
- What is (2/5)2?
- Simplify x2 × x2.
- Simplify 4x2 × 5x2.
- A square rug has an area of 81 square feet. How long is each side?
- Expand (x + 5)2.
Answers
- 7 × 7 = 49.
- 12 × 12 = 144.
- (-6)2 = 36. -62 = -36.
- (2 × 2)/(5 × 5) = 4/25.
- x4.
- (4 × 5)x2+2 = 20x4.
- √81 = 9 feet.
- (x + 5)(x + 5) = x2 + 5x + 5x + 25 = x2 + 10x + 25.
If your kid is stuck on squared numbers
Squaring is just multiplication, so trouble here usually means trouble one step back. A kid who can’t find 72 quickly probably isn’t fluent with the times tables, which should be automatic by the end of third grade or so. A kid who’s fine with 72 but lost on x2 × x2 often never got a clear picture of what an exponent counts, so the rules feel like random tricks to memorize.
Our math skills by grade level guide shows when multiplication facts should be locked in. For older kids tangled in exponent rules, Struggling With Algebra? The Real Problem Is Usually Fourth Grade Fractions explains how early gaps surface in algebra. And since math facts stick best with frequent short sessions, see Why 20 Minutes A Day Beats A 90-Minute Tutoring Session.
Frequently asked questions
What does squared mean in math?
It means multiplying a number by itself. 9 squared is 9 × 9 = 81.
What is squared times squared?
You add the exponents, so x2 × x2 = x4. With numbers, 52 × 52 = 25 × 25 = 625, which equals 54.
Is squared the same as times 2?
No. Squared means times itself. 10 squared is 100, while 10 times 2 is 20. The only numbers where the two match are 0 and 2.
What is x squared plus x squared?
2x2. You’re adding two of the same thing, so the exponent doesn’t change.
Can a squared number be negative?
Not if the number is a real number. Any real number squared is zero or positive. An expression like -42 is negative only because the minus sign is applied after squaring.
How do you type squared?
On most calculators and in spreadsheets, use the ^ key: 5^2 means 5 squared. In many word processors you can type 5 and a 2, then format the 2 as superscript.
Find the actual gap
If squared numbers or exponent rules keep tripping up homework, it’s worth knowing whether the real issue is multiplication facts, negatives or something else. Math Anon’s short adaptive assessment checks those underlying skills and gives you a written report showing which ones are missing.
Your kid then works on exactly those skills for about 20 minutes a day, and you get a progress report every week. The assessment is $49 and gets credited back in full on the first month. Start the assessment or look at pricing.
