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Square Root Rules — Definition, Formulas & Examples

Square Root Rules

Algebra rules for square roots are listed below. Square root rules are a subset of nth root rules and exponent rules.

 

Definitions

1. Math formula: b = √a, defining b as the square root of a if both b ≥ 0 and b2 = a.

2. Math formula showing the square root of a squared equals the absolute value of a: √(a²) = |a|

Examples

Math equation showing square root of 9 equals 3 because 32 = 9.

Square root of (-5)² equals |-5| equals 5, showing absolute value is needed when variable could be negative.

3. If a ≥ 0 then Square root of a squared equals a, i.e., √(a²) = a.

 

Square root of 12 squared equals 12, showing that √(12²) = 12

Distributing (a ≥ 0 and b ≥ 0)

1. Square root of (ab) equals square root of a times square root of b

2. Square root rule: sqrt(a/b) = sqrt(a) / sqrt(b)     (b ≠ 0)

3. Math formula showing square root rule: √a × √a = a

Examples

Math equation showing √18 = √(9·2) = √9·√2 = 3√2, demonstrating the distributive property of square roots.

Square root of (9/4) equals square root of 9 divided by square root of 4, which equals 3/2

Square root rule: √10 · √10 = 10, showing that multiplying a square root by itself equals the original number.

4. Square root of a^n equals (square root of a)^n, or equivalently a^(n/2)

 

Math equation showing √(2^6) = 2^(6/2) = 2^3 = 8, demonstrating the square root of an exponential expression.
Rationalizing the Denominator
(a > 0, b > 0, c > 0) 		Examples

Math equation showing step 1: a/√b = a/√b · √b/√b = a√b/b, demonstrating rationalization by multiplying by √b/√b.

Math example showing a/(b+√c) rationalized by multiplying by (b−√c)/(b−√c) to get (ab−a√c)/(b²−c)

Equation showing rationalization: 4/√2 = (4/√2)·(√2/√2) = 4√2/2 = 2√2

Step-by-step rationalization of 6/(3+√7) by multiplying by (3−√7)/(3−√7), yielding (18−6√7)/2 = 9−3√7.
Math example 3: a/(b−√c) = a/(b−√c) · (b+√c)/(b+√c) = (ab+a√c)/(b²−c) Step-by-step rationalization of 5/(2−√3) by multiplying by (2+√3)/(2+√3), simplifying to 10+5√3.
Math example showing a/(√b + √c) rationalized by multiplying by (√b − √c)/(√b − √c) = (a√b − a√c)/(b − c) Rationalizing 14/(√13+√11) by multiplying by (√13−√11)/(√13−√11), simplifying to (14√13−14√11)/2 = 7√13−7√11.
Math example showing: a/(√b−√c) = a(√b+√c)/(b−c), rationalizing by multiplying by (√b+√c)/(√b+√c).

Rationalizing 12/(√15−√7) by multiplying by (√15+√7)/(√15+√7), simplifying to (3√15+3√7)/2.

 

Careful!!

1. Math warning: square root of (a + b) does NOT equal square root of a plus square root of b

2. Math warning: square root of (a minus b) does NOT equal square root of a minus square root of b

3. Math warning: square root of (a² + b²) does not equal a + b

Examples

Math warning showing √(6+10) ≠ √6 + √10, illustrating that square roots cannot be distributed over addition.

Math warning: sqrt(6 − 2) ≠ sqrt(6) − sqrt(2); subtraction cannot be distributed under a square root.

Math warning: square root of (9 + 4) ≠ 3 + 2, shown in yellow, illustrating that square roots do not distribute over addition.

 

See also

nth root, radical, factoring rules

Key Formula

a⋅b=a⋅bab=ab(a)2=a\sqrt{a \cdot b} = \sqrt{a}\cdot\sqrt{b} \qquad \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \qquad \left(\sqrt{a}\right)^2 = a
Where:
  • aa = A non-negative real number under the radical
  • bb = A positive real number (positive when in a denominator)

Worked Example

Problem: Simplify the expression: √50 + 3√8 − √18.
Step 1: Factor each radicand to extract perfect squares. Start with √50.
50=25⋅2=25⋅2=52\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25}\cdot\sqrt{2} = 5\sqrt{2}
Step 2: Simplify 3√8 by factoring 8 as 4 × 2.
38=34⋅2=3⋅22=623\sqrt{8} = 3\sqrt{4 \cdot 2} = 3 \cdot 2\sqrt{2} = 6\sqrt{2}
Step 3: Simplify √18 by factoring 18 as 9 × 2.
18=9⋅2=32\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}
Step 4: Combine like terms — all three terms now contain √2.
52+62−32=(5+6−3)2=825\sqrt{2} + 6\sqrt{2} - 3\sqrt{2} = (5 + 6 - 3)\sqrt{2} = 8\sqrt{2}
Answer: 8√2

Another Example

Problem: Rationalize the denominator of 6 / √3.
Step 1: Multiply both the numerator and denominator by √3 so the denominator becomes a rational number.
63⋅33=63(3)2\frac{6}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{\left(\sqrt{3}\right)^2}
Step 2: Simplify the denominator using the rule (√a)² = a.
633\frac{6\sqrt{3}}{3}
Step 3: Reduce the fraction.
633=23\frac{6\sqrt{3}}{3} = 2\sqrt{3}
Answer: 2√3

Frequently Asked Questions

Can you add or subtract square roots directly?
You can only add or subtract square roots that have the same radicand (the number under the radical). For example, 2√5 + 7√5 = 9√5, but √2 + √3 cannot be combined into a single term. This is similar to how you can combine like terms in algebra but not unlike terms.
Does the square root of a sum equal the sum of the square roots?
No. A very common error is writing √(a + b) = √a + √b, but this is false. For instance, √(9 + 16) = √25 = 5, while √9 + √16 = 3 + 4 = 7. The product rule √(ab) = √a · √b works, but there is no corresponding rule for addition.

Why It Matters

Square root rules are essential for simplifying expressions throughout algebra, geometry, and beyond. You use them when solving quadratic equations, working with the Pythagorean theorem, and simplifying distance formulas. Mastering these rules also prepares you for working with higher-index radicals and rational exponents in precalculus.

Common Mistakes

Mistake: Writing √(a + b) = √a + √b (distributing the square root over addition).
Correction: The square root distributes over multiplication, not addition. √(a · b) = √a · √b is valid, but √(a + b) must stay as is unless you can factor the expression differently. Test with numbers: √(4 + 9) = √13 ≈ 3.6, not √4 + √9 = 5.
Mistake: Forgetting that √(x²) = |x|, not simply x.
Correction: The square root function returns a non-negative value. If x could be negative, then √(x²) = |x|. For example, √((-5)²) = √25 = 5, not −5. Only when you know x ≥ 0 can you write √(x²) = x.

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