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Square Root Rules

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