What Is Squaring a Negative Number? (Math Rules)

Squaring a negative number means multiplying it by itself. A negative times a negative is positive, so ((-4)^2=(-4)(-4)=16). Parentheses matter: they show that the negative number is the base. Without parentheses, (-4^2) means the opposite of (4^2), which is (-16).

You may meet this rule while working through homework, checking a calculator, or trying to understand an equation. The main source of confusion is often not the multiplication. It is knowing which part of the expression gets squared. Once you spot the base and keep its sign attached, the steps are simple.

What Squaring a Negative Number Means

Squaring means multiplying a number by itself. If the number is negative, both factors are negative, and their product is positive. For any positive real number (a), ((-a)^2=(-a)(-a)=a^2). Zero squared is zero, so every real number squared is nonnegative.

For example, to square negative four, write the operation as repeated multiplication:

[ (-4)^2=(-4)\times(-4)=16 ]

The two negative signs belong to the two factors. The multiplication rule says that a negative times a negative gives a positive result. Then multiply the number parts: (4\times4=16).

A useful way to remember this is that a square uses two matching factors. The sign rule applies to the pair, while the number itself is multiplied by itself. Here are a few examples:

A negative number and its opposite have the same square. For example, ((-6)^2=36), and (6^2=36). The inputs have different signs, but each is multiplied by itself, so the final products match.

This rule also gives you a quick check: the square of a real number cannot be negative. If your answer to ((-7)^2) is (-49), go back and check how you handled the two negative factors.

Parentheses, Order of Operations, and Worked Examples

Parentheses identify the base, or the quantity being raised to a power. In ((-3)^2), the base is negative three; in (-3^2), the base is positive three, and a negative sign sits outside the power. Under standard order of operations, these expressions have different values.

The small raised number is called an exponent. In (3^2), the exponent is (2), and it tells you to use two factors of the base: (3\times3). Parentheses can bring a negative sign into that base.

Compare the two expressions:

[ (-3)^2=(-3)(-3)=9 ]

[ -3^2=-(3^2)=-(3\times3)=-9 ]

In the first expression, the parentheses make the entire number (-3) the base. In the second, the exponent applies to (3), and the minus sign is applied after that power is found.

A reliable method is to pause before calculating:

• Find the exponent.
• Identify the base. Check whether the negative sign is inside parentheses.
• Rewrite the square as two matching factors.
• Multiply the signs, then multiply the number parts.
• Check that a real number squared is not negative.

Try the method with ((-8)^2). The base is (-8), so write ((-8)(-8)). The signs make a positive, and (8\times8=64). Thus, ((-8)^2=64).

Now try (-8^2). There are no parentheses around (-8), so the exponent applies to (8) first: (-(8\times8)=-64). The location of the minus sign changes the answer.

Common Errors and How to Check Your Answer

The most common mistake is dropping parentheses or acting as if a minus sign always belongs to the number next to it. Another is thinking that every expression with a negative sign must have a negative value. Use the written base, the multiplication steps, and a quick sign check to catch these errors.

Error: Treating (-3^2) as ((-3)^2)

These expressions look similar, but they are not equal. In (-3^2), standard order of operations says to calculate the exponent first, then apply the negative sign. So (-3^2=-(3\times3)=-9). In ((-3)^2), the negative number is the base, so the answer is (9).

Error: Making a negative square

If the whole negative number is squared, show both factors before you decide on the sign:

[ (-6)^2=(-6)(-6)=36 ]

Two negative factors give a positive product. If you write ((-6)^2=-36), you have likely treated the expression as (-(6^2)), which is different.

Error: Squaring only the number part and forgetting the sign

For ((-9)^2), the base is the full number (-9), not just (9). Write the repeated multiplication as ((-9)(-9)). Keeping the parentheses in this step helps you carry the negative sign through the calculation.

A three-part answer check

After solving, ask:

• Did I identify the base correctly? If the negative sign is inside parentheses, it is part of the base.
• Did I use two copies of the base? A square has two matching factors.
• Does the sign make sense? A real number squared is positive unless the number is zero, in which case its square is zero.

That last check is a diagnostic, not a replacement for the steps. It can show that an answer is wrong, but it cannot tell you whether you misplaced parentheses or made a multiplication error.

Worked Scenarios: Applying the Rule

A worked scenario shows how the rule holds up when you use it in a longer calculation. These examples stay focused on signed numbers and powers. In each one, first decide what is being squared, then multiply the factors and check the result.

Scenario 1: Squaring a negative coordinate

On a number line, a point may be described by a signed coordinate such as (-5). If a calculation asks for the square of that coordinate, write ((-5)^2), because the entire coordinate is the base:

[ (-5)^2=(-5)(-5)=25 ]

The square is (25), not (-25). The coordinate itself remains negative; squaring it produces a positive value.

Scenario 2: Comparing opposite inputs

Suppose you need to compare the squares of (7) and (-7). Calculate each with its own sign:

[ 7^2=7\times7=49 ]

[ (-7)^2=(-7)\times(-7)=49 ]

The results are equal because a number and its opposite have the same square. This does not mean the original numbers are equal: (7) and (-7) are distinct inputs.

Scenario 3: Checking a multi-step expression

Consider (2+(-3)^2). The parentheses show that (-3) is squared. Find the power first, then add:

[ 2+(-3)^2=2+9=11 ]

If the expression were (2-3^2), the exponent would apply to (3), giving (2-9=-7). A small change in the expression changes the result, so keep the original notation in view.

These examples use the same practical habit: do not calculate from appearance alone. Mark the base, rewrite the power, and then use the sign rule.

Where Squaring Appears Next: Equations and Graphs

The square of a number appears in equations and graphs. The key sign rule remains the same: squaring a negative input gives a positive output. Keeping that distinction clear helps you read expressions such as (x^2), solve simple equations, and understand why opposite inputs can produce matching values.

In an equation such as (x^2=25), the question is: which real numbers have a square of (25)? Both (5) and (-5) work:

[ 5^2=25 \qquad\text{and}\qquad (-5)^2=25 ]

So the real solutions are (x=5) and (x=-5). Do not confuse this with (\sqrt{25}). The square-root symbol (\sqrt{25}) refers to the nonnegative value (5); solving (x^2=25) asks for all real values of (x) that work.

In a graph of (y=x^2), positive and negative inputs with the same distance from zero have the same output. For example, (x=4) and (x=-4) both give (y=16). The graph therefore has matching heights at those two inputs.

Key takeaways

• Squaring means using a number as both factors: (x^2=x\times x).
• A negative base squared gives a positive result: ((-a)^2=a^2) for (a>0).
• Zero squared is zero, so every real square is nonnegative.
• Parentheses decide whether the negative sign is part of the base.
• Under standard order of operations, (-3^2=-9), but ((-3)^2=9).

Frequently Asked Questions

Is the square of a negative number positive?

Yes. When a negative number is the base, squaring multiplies two negative factors. Their product is positive. For example, ((-4)^2=(-4)(-4)=16).

Why does a negative times a negative make a positive?

This is a standard rule for multiplying signed numbers. When you square a negative number, the rule applies to its two matching negative factors, making the product positive.

What is ((-5)^2)?

((-5)^2=25). The parentheses make (-5) the base, so calculate ((-5)(-5)).

What is (-5^2)?

Under standard order of operations, (-5^2=-(5^2)=-25). The exponent applies to (5), and the negative sign is outside the power.

Do parentheses change the answer when squaring a negative number?

They can. ((-5)^2=25), while (-5^2=-25). Parentheses show whether the negative sign is part of the base.

Can the square of a real number be negative?

No. The square of any real number is nonnegative. A nonzero real number has a positive square, and zero has a square of zero.

Does a negative number and its opposite have the same square?

Yes. A number and its opposite have the same square. For example, ((-8)^2=64) and (8^2=64).

What is zero squared?

Zero squared is zero: (0^2=0\times0=0).

How can I check whether I squared a negative number correctly?

Check that the negative sign is part of the base, rewrite the expression as two factors, and multiply. If the whole negative number is squared, the result should not be negative.

Is (\sqrt{25}) the same as solving (x^2=25)?

No. The symbol (\sqrt{25}) means the nonnegative square root, (5). The equation (x^2=25) has two real solutions: (5) and (-5).

(This article was written by one of our staff writers, Nathan Sterling. Visit our Meet the Team page.)