Square Root of a Square- Finding It Correctly

Why Everyone Gets Confused About √(x²)

Here's the thing: most students see √(x²) and immediately write x. They're wrong. The correct answer is |x|—the absolute value of x. This isn't a technicality or a trick. It's the actual mathematical truth.

The confusion starts because teachers often use positive numbers in examples. When x = 4, √(16) = 4. That works. But plug in x = -3 and you get √(9) = 3, not -3. That's where the absolute value comes in.

The Hard Rule: √(x²) = |x|

When you square a number, you lose the sign. When you take the square root, you only get the principal (non-negative) root. So if x is negative, you need to account for that lost sign.

• If x ≥ 0, then √(x²) = x
• If x < 0, then √(x²) = -x (which equals |x|)

That's it. No exceptions, no edge cases. Just remember: squaring erases the sign, square roots don't bring it back.

Step-by-Step: Finding √(x²) Correctly

Method 1: Plug in a Number

If you have a specific value for x, just compute it directly.

Example: Find √(7²)

1. Calculate 7² = 49
2. Find √49
3. Answer: 7

Example: Find √((-5)²)

1. Calculate (-5)² = 25
2. Find √25
3. Answer: 5

Method 2: Keep x as a Variable

If x is unknown, you have to keep the absolute value form.

Example: Simplify √(x²)

Answer: |x|

You cannot simplify further without knowing whether x is positive or negative.

Common Mistakes That Will Cost You Points

• Writing √(x²) = x: Only valid when x ≥ 0. Wrong for negative x.
• Confusing √(x²) with (√x)²: These are different. (√x)² = x (for x ≥ 0), but √(x²) = |x|.
• Dropping absolute value symbols: If your answer sheet says |x|, keep it. Your teacher will mark it wrong otherwise.
• Ignoring the domain: √x only exists for x ≥ 0. This affects (√x)² differently than √(x²).

Side-by-Side Comparison

Quick Examples With Numbers

1. x = 9
√(9²) = √81 = 9 ✓

2. x = -9
√((-9)²) = √81 = 9 = |-9| ✓

3. x = 0
√(0²) = √0 = 0 = |0| ✓

4. x = ½
√((½)²) = √(¼) = ½ = |½| ✓

The pattern holds every time. Zero works. Fractions work. Negative numbers work. The only thing that changes is how you write the answer.

When Teachers Ask for "The Square Root of a Square"

Sometimes the question phrasing is vague. If they ask for "the square root of a²," they might accept both |a| and ±a depending on context. But in strict mathematical terms:

• √(a²) = |a|
• ±√(a²) = ±|a| (which gives you both a and -a)

The ± version is technically incorrect notation since √ already means principal root. But some textbooks use it loosely. Know what your instructor expects.

The Bottom Line

√(x²) is always the absolute value of x. There's no shortcut, no trick, no exception. Memorize it, practice it, and stop overcomplicating it.

When in doubt: square first, take the root second, check your sign. If the original x was negative, your answer is positive. That's the whole game.