Why does √9 equal 3 and not −3?

The radical symbol √ denotes the principal (non-negative) square root. The equation x² = 9 has two solutions: 3 and −3. But √9 alone is defined as 3. If you need the negative root, write −√9 = −3.

Can I multiply square roots of negative numbers?

Not with the product property for real numbers. The rule √(ab) = √a × √b applies only when a and b are non-negative. For negative radicands, you must first convert to imaginary numbers: √(−4) = 2i. Then multiply: 2i × 3i = 6i² = −6. But √((−4) × (−9)) = √36 = 6, which is different. Always check the sign of the radicand.

How do I add √2 and √3?

You cannot combine them into a single term. √2 and √3 are not like radicals because their radicands differ. The sum remains √2 + √3. You can evaluate it approximately: 1.414 + 1.732 = 3.146, but the exact expression is √2 + √3.

What does rationalising the denominator achieve?

It removes a radical from the denominator of a fraction. The result is in simplest radical form, which is the conventional way to present answers. For instance, rationalising 1/√2 gives √2/2. The value is the same, but the expression is standardised.

When do I use the ± symbol with a square root?

The ± symbol appears when solving an equation like x² = 9. The solution is x = ±3, meaning both 3 and −3 satisfy the equation. The radical symbol √ alone never carries the ±; it always gives the principal root.

Square Root Rules: Product, Quotient and Exponents

Square Root Formula and Rules

Most students first encounter the square root symbol √ and assume that √9 equals ±3. It does not. The symbol √9 alone means the principal, non-negative square root, which is 3. The negative root is written as −√9, which is −3. The square root formula √x = y defines y as the value that satisfies y² = x and y ≥ 0, so for real numbers that value is always non-negative when you see the radical sign. This distinction is the foundation of every rule that follows.

Every positive number has two square roots: one positive and one negative. The principal square root is the non-negative one. The square root of zero is zero. The square root of a negative number is not a real number; it is defined in the complex number system using the imaginary unit i, where √(−1) = i. All radicands here are non-negative unless stated otherwise.

Square Root Rules Reference Table

Core Properties of Square Roots

The following properties govern how square roots behave under multiplication, division, exponentiation, and addition. These are the rules you apply when simplifying expressions and solving equations.

Product Property: For non-negative a and b, √(a × b) = √a × √b. Instance: √(4 × 25) = √4 × √25 = 2 × 5 = 10.

Quotient Property: For non-negative a and positive b, √(a ÷ b) = √a ÷ √b. Instance: √(16 ÷ 4) = √16 ÷ √4 = 4 ÷ 2 = 2.

Square of a Square Root: (√a)² = a for a ≥ 0. Instance: (√7)² = 7.

Square Root of a Square: √(a²) = |a|. The absolute value ensures the result is non-negative. Instance: √((−5)²) = √25 = 5 = |−5|. This is a common failure point, never write √(a²) = a.

Square Root as an Exponent: √a = a^(1/2) for a ≥ 0. This means you can use all exponent rules on square roots. Instance: √a × √a = a^(1/2) × a^(1/2) = a¹ = a.

Adding and Subtracting Like Radicals: You can combine radicals only when they have the same index and the same radicand. Instance: 3√5 + 2√5 = 5√5. Instance: √2 + √3 cannot be combined.

Multiplying Radicals: Use the product property, then simplify. Instance: √6 × √10 = √60 = √(4 × 15) = 2√15.

Dividing Radicals: Use the quotient property, then simplify. Instance: √18 ÷ √2 = √9 = 3.

These rules are covered in OpenStax Intermediate Algebra 2e, sections 8.2 through 8.5. Consult those sections for the formal statements and additional practice.

Product Property of Square Roots

How to Multiply and Simplify

The product property states that for non-negative a and b, √(ab) = √a × √b. This is the rule you use when multiplying square roots and when simplifying radicals by factoring out perfect squares.

Worked Instance: Multiply √8 × √2
√(8 × 2) = √16 = 4. Alternatively, you could simplify first: √8 = 2√2, then 2√2 × √2 = 2 × 2 = 4. Both approaches give the same result.

Worked Instance: Simplify √72
Factor the radicand: 72 = 36 × 2. Then √72 = √(36 × 2) = √36 × √2 = 6√2. This is simplest radical form, no perfect square factor remains inside the radical.

Common Mistake: Applying the product property when a or b is negative. The property holds only for non-negative radicands. For instance, √((−4) × (−9)) = √36 = 6, but √(−4) × √(−9) = 2i × 3i = 6i² = −6, which is wrong. Stick to non-negative radicands in real-number work.

Quotient Property of Square Roots

How to Divide and Rationalise

For non-negative a and positive b, √(a ÷ b) = √a ÷ √b. This property is used when dividing square roots and when simplifying fractions under a radical.

Worked Instance: Divide √45 by √5
√45 ÷ √5 = √(45 ÷ 5) = √9 = 3. Always check that the radicand under division yields a whole number where possible.

Worked Instance: Simplify √(25/9)
√(25 ÷ 9) = √25 ÷ √9 = 5 ÷ 3 = 5/3. This works because the quotient property applies exactly the same way to fractions.

Rationalising the Denominator: When a radical appears in the denominator of a fraction, you eliminate it by multiplying numerator and denominator by the radical. For 1 ÷ √2, multiply by √2 ÷ √2 to get √2 ÷ 2. The denominator is now rational. This is a required step for simplest radical form: no radical in the denominator.

Worked Instance: Rationalise 5 ÷ √3
Multiply: (5 × √3) ÷ (√3 × √3) = 5√3 ÷ 3. The quotient property does not directly apply here because the denominator contains a radical; rationalising is the correct procedure.

Square Root as Exponent

Using Exponent Rules with Roots

The square root of a number is equivalent to raising that number to the power of 1/2. The formula √a = a^(1/2) works for any non-negative a. This relationship means that every exponent rule, product rule, quotient rule, power rule, applies to square roots.

Worked Instance: Write √x × √x using exponents
√x × √x = x^(1/2) × x^(1/2) = x^(1/2 + 1/2) = x¹ = x. This matches the rule that (√x)² = x.

Worked Instance: Simplify (√a)³
Write √a as a^(1/2). Then (a^(1/2))³ = a^(3/2). In radical form, a^(3/2) = (√(a³)) only for non-negative a. This is how you move between radical and exponential notation.

Spreadsheet Use: Most spreadsheet programs have a SQRT function for square roots. You can also use the POWER function: POWER(x, 0.5) returns √x. Both methods are built into Google Sheets and Microsoft Excel. For cube roots or higher, use POWER with exponent 1/3 or 1/n.

Adding and Subtracting Like Radicals

Combining Like Terms Under a Root

You can add or subtract square roots only when they are like radicals, meaning they have the same radicand. The coefficient in front of the radical is what you combine; the radical part stays unchanged.

Worked Instance: Add 4√6 + 7√6
Both terms have radicand 6, so you add the coefficients: 4 + 7 = 11, giving 11√6.

Worked Instance: Subtract 9√3 − 2√3
9 − 2 = 7, so the result is 7√3.

What You Cannot Do: √(a + b) does not equal √a + √b. This is a frequent error. For instance, √(9 + 16) = √25 = 5, not 3 + 4 = 7. There is no distributive property for addition under a square root. If the radicand is not a sum of perfect squares that can be evaluated first, you cannot simplify √(a + b) by separating the terms.

Rationalising the Denominator

Removing Radicals from the Bottom

A radical expression is in simplest radical form when no radical appears in the denominator. To remove a radical from the denominator, multiply both the numerator and the denominator by the radical that appears. This process is called rationalising the denominator.

Worked Instance: Rationalise 3 ÷ √7
Multiply numerator and denominator by √7: (3 × √7) ÷ (√7 × √7) = 3√7 ÷ 7. The denominator is now 7, a rational number.

Worked Instance: Rationalise (2 + √5) ÷ √3
Multiply by √3: (2√3 + √15) ÷ 3. The denominator is rationalised.

Using Conjugates: When the denominator has the form a + √b, multiply numerator and denominator by the conjugate a − √b. The product of conjugates is a² − b, which is rational. Instance: rationalise 1 ÷ (3 + √2). Multiply by (3 − √2): (3 − √2) ÷ (9 − 2) = (3 − √2) ÷ 7.

This technique is covered in OpenStax Intermediate Algebra 2e, section 8.4. You need it to simplify expressions in algebra and calculus.

What You Cannot Do With Square Roots

Several square root rules are tempting but false. Knowing these pitfalls prevents algebra errors.

√(a + b) ≠ √a + √b. This is the most common mistake. The square root of a sum is not the sum of the square roots. There is no distributive property. Instance: √(25 + 144) = √169 = 13, not 5 + 12 = 17.

√(a − b) ≠ √a − √b. Same reason. Instance: √(100 − 36) = √64 = 8, not 10 − 6 = 4.

√(−a) is not a real number. For real-number work, the radicand must be non-negative. The square root of a negative number exists only in the complex number system, where √(−a) = i√a. If you try to take the square root of a negative number on a standard calculator, you will get an error.

√(a²) ≠ a. The correct result is |a|, the absolute value. For a = −3, √((−3)²) = √9 = 3, which is |−3|, not −3.

You cannot simplify √(a) + √(b) unless a = b. Two square roots with different radicands cannot be combined into a single radical.