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Operations on Roots



Key Terms
Like Radicals
Adding and Subtracting Like Radicals
Multiplying Radicals with the Same Index
Multiplying Radicals with Different Indices
Dividing Radicals
Conjugates
Expanding Products with Radicals
Squaring Radical Expressions
Operations Strategy
Operations on Roots FAQ



Combining Radical Expressions

Radicals can be added, subtracted, multiplied, and divided — but each operation has its own requirements. Addition and subtraction work only with like radicals. Multiplication and division follow the radical rules but require matching indices.

Mastering these operations transforms complicated expressions into simpler ones and prepares the ground for solving equations where radicals appear.

Key Terms

Combining Radicals

Like Radicals— radicals with the same index and radicand; only like radicals can be added or subtracted
Conjugate— a binomial with the sign between terms reversed; its product with the original eliminates the radical

Rules Used

Product Rule (Radicals)— multiplying radicals with the same index
Quotient Rule (Radicals)— dividing radicals with the same index
Rational Exponent— used to multiply radicals with different indices by finding a common denominator
Simplest Form— simplify each radical before attempting to add or subtract

See All Algebra Definitions →


Like Radicals

Like radicals share the same index and the same radicand. Only like radicals can be added or subtracted.

35 and 75are like radicals3\sqrt{5} \text{ and } 7\sqrt{5} \quad \text{are like radicals}


3 and 5are not like — different radicands\sqrt{3} \text{ and } \sqrt{5} \quad \text{are not like — different radicands}


2 and 23are not like — different indices\sqrt{2} \text{ and } \sqrt[3]{2} \quad \text{are not like — different indices}


The concept parallels like terms in algebra. Just as 3x3x and 7x7x can combine but 3x3x and 7y7y cannot, radicals combine only when their radical parts match exactly.

Recognizing like radicals sometimes requires simplification first. Two radicals that appear different may become like after simplifying.
Pair Same index? Same radicand? Like radicals?
3√5,   7√5 ✓ ✓ ✓
√3,   √5 ✓ ✗ ✗
√2,   ∛2 ✗ ✓ ✗

Adding and Subtracting Like Radicals

To add or subtract like radicals, combine the coefficients and keep the radical unchanged.

35+75=1053\sqrt{5} + 7\sqrt{5} = 10\sqrt{5}


823−323=5238\sqrt[3]{2} - 3\sqrt[3]{2} = 5\sqrt[3]{2}


127+347=547\frac{1}{2}\sqrt{7} + \frac{3}{4}\sqrt{7} = \frac{5}{4}\sqrt{7}


Unlike radicals cannot be combined:

2+3\sqrt{2} + \sqrt{3}


This expression is already in simplest form. No further reduction is possible.

When radicals appear unlike but can be simplified to like form:

12+27=23+33=53\sqrt{12} + \sqrt{27} = 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}


Always simplify each radical before attempting to add or subtract.

Multiplying Radicals with the Same Index

When radicals share the same index, multiply the radicands and keep the index.

a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}


This is the product rule from radical rules.

3⋅12=36=6\sqrt{3} \cdot \sqrt{12} = \sqrt{36} = 6


43⋅23=83=2\sqrt[3]{4} \cdot \sqrt[3]{2} = \sqrt[3]{8} = 2


When coefficients are present, multiply them separately:

32⋅45=12103\sqrt{2} \cdot 4\sqrt{5} = 12\sqrt{10}


(23)2=22⋅(3)2=4⋅3=12(2\sqrt{3})^2 = 2^2 \cdot (\sqrt{3})^2 = 4 \cdot 3 = 12


Multiplication often produces results that can be simplified:

6⋅8=48=16⋅3=43\sqrt{6} \cdot \sqrt{8} = \sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}

Multiplying Radicals with Different Indices

When indices differ, the product rule does not apply directly. Convert to rational exponents, find a common denominator, then proceed.

2⋅23=21/2⋅21/3\sqrt{2} \cdot \sqrt[3]{2} = 2^{1/2} \cdot 2^{1/3}


Find a common denominator for the exponents:

=23/6⋅22/6=25/6=256=326= 2^{3/6} \cdot 2^{2/6} = 2^{5/6} = \sqrt[6]{2^5} = \sqrt[6]{32}


Alternatively, convert both radicals to the same index first:

2=236=86\sqrt{2} = \sqrt[6]{2^3} = \sqrt[6]{8}


23=226=46\sqrt[3]{2} = \sqrt[6]{2^2} = \sqrt[6]{4}


86⋅46=326\sqrt[6]{8} \cdot \sqrt[6]{4} = \sqrt[6]{32}


Both methods reach the same result. The exponent approach is often more efficient for complex expressions.

Dividing Radicals

Division follows the quotient rule from radical rules:

ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}


502=502=25=5\frac{\sqrt{50}}{\sqrt{2}} = \sqrt{\frac{50}{2}} = \sqrt{25} = 5


54323=5423=273=3\frac{\sqrt[3]{54}}{\sqrt[3]{2}} = \sqrt[3]{\frac{54}{2}} = \sqrt[3]{27} = 3


With coefficients:

121543=124⋅153=35\frac{12\sqrt{15}}{4\sqrt{3}} = \frac{12}{4} \cdot \sqrt{\frac{15}{3}} = 3\sqrt{5}


When indices differ, convert to rational exponents:

843=23/222/3=23/2−2/3=29/6−4/6=25/6=326\frac{\sqrt{8}}{\sqrt[3]{4}} = \frac{2^{3/2}}{2^{2/3}} = 2^{3/2 - 2/3} = 2^{9/6 - 4/6} = 2^{5/6} = \sqrt[6]{32}


If the result leaves a radical in the denominator, rationalize to achieve simplest form.

Conjugates

The conjugate of a binomial expression reverses the sign between terms.

Conjugate of a+ba + \sqrt{b} is a−ba - \sqrt{b}.

Conjugate of a−b\sqrt{a} - \sqrt{b} is a+b\sqrt{a} + \sqrt{b}.

The key property: multiplying conjugates eliminates radicals.

(a+b)(a−b)=a2−b(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b


(a+b)(a−b)=a−b(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b


These are difference of squares patterns.

(3+2)(3−2)=9−2=7( 3 + \sqrt{2})(3 - \sqrt{2}) = 9 - 2 = 7


(7+3)(7−3)=7−3=4(\sqrt{7} + \sqrt{3})(\sqrt{7} - \sqrt{3}) = 7 - 3 = 4


Conjugates are essential for rationalizing denominators and appear in solving certain radical equations.
Pattern Conjugate Product (radical eliminated) Numerical example
a + √b a − √b a² − b (3 + √2)(3 − √2) = 7
√a + √b √a − √b a − b (√7 + √3)(√7 − √3) = 4

Expanding Products with Radicals

Binomials containing radicals expand using the distributive property, just like algebraic binomials.

(2+3)(4−3)(2 + \sqrt{3})(4 - \sqrt{3})


Apply FOIL:

=2⋅4+2⋅(−3)+3⋅4+3⋅(−3)= 2 \cdot 4 + 2 \cdot (-\sqrt{3}) + \sqrt{3} \cdot 4 + \sqrt{3} \cdot (-\sqrt{3})


=8−23+43−3= 8 - 2\sqrt{3} + 4\sqrt{3} - 3


=5+23= 5 + 2\sqrt{3}


Squaring a binomial:

(5+2)2=(5)2+252+(2)2(\sqrt{5} + \sqrt{2})^2 = (\sqrt{5})^2 + 2\sqrt{5}\sqrt{2} + (\sqrt{2})^2


=5+210+2=7+210= 5 + 2\sqrt{10} + 2 = 7 + 2\sqrt{10}


Common error: (a+b)2≠a2+b(a + \sqrt{b})^2 \neq a^2 + b. The middle term 2ab2a\sqrt{b} must not be forgotten.

These expansions appear frequently in radical equations when both sides are squared.
a²b²ababa + ba + babab
Why the middle term stays

The square of a sum as an area: a side a + b cut into an a × a square, a b × b square and two a × b rectangles. With a = √5 and b = √2 the squares have areas 5 and 2 and each rectangle has area √5·√2 = √10, so (√5 + √2)² = 7 + 2√10. Writing a² + b² alone drops the two rectangles. Build the square piece by piece on the square of a sum explorer.

The two rectangles are the 2a√b of (a + √b)², which is why squaring a binomial leaves a radical behind.

Squaring Radical Expressions

Squaring eliminates square roots:

(x)2=x(\sqrt{x})^2 = x


(35)2=9⋅5=45(3\sqrt{5})^2 = 9 \cdot 5 = 45


For binomials, expand fully:

(x+3)2=x+6x+9(\sqrt{x} + 3)^2 = x + 6\sqrt{x} + 9


(x+1−2)2=(x+1)−4x+1+4=x+5−4x+1(\sqrt{x + 1} - 2)^2 = (x + 1) - 4\sqrt{x + 1} + 4 = x + 5 - 4\sqrt{x + 1}


When solving radical equations, squaring both sides is the primary technique for eliminating radicals. The expressions above show what results from squaring — and why squaring a binomial with a radical does not fully eliminate the radical unless conjugates are involved.

Squaring can also introduce extraneous solutions. Any solution must be checked in the original equation, as detailed in the equations page.

Operations Strategy

A systematic approach to radical operations:

Before adding or subtracting, simplify each radical to identify like radicals.

Before multiplying or dividing, check whether indices match. If they do, apply the product or quotient rule. If not, convert to rational exponents.

After any operation, simplify the result. Multiplication often creates perfect power factors. Division may leave radicals in denominators requiring rationalization.

When expanding products, apply distributive property carefully. Track each term. Combine like radicals at the end.

These operations prepare expressions for solving radical equations and for analyzing radical functions. Fluency here makes those topics more accessible.
Operation Requirement Rule Example
Add / subtract like radicals (same index AND same radicand) combine coefficients, keep the radical unchanged 3√5 + 7√5 = 10√5
Multiply same index — else convert to rational exponents product rule; multiply coefficients separately √3 · √12 = 6
Divide same index — else convert to rational exponents quotient rule; rationalize if a radical remains √50 ⁄ √2 = 5
Expand binomial product none distributive (FOIL); combine like radicals at the end (2 + √3)(4 − √3) = 5 + 2√3
Square a binomial none (a + b)² formula — keep the middle term 2ab (√5 + √2)² = 7 + 2√10
Multiply conjugates binomial × its conjugate difference of squares — the radicals eliminate (a + √b)(a − √b) = a² − b

Operations on Roots FAQ

Why can't you add 2+3\sqrt{2} + \sqrt{3}?

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Because they are unlike radicals, exactly as 3x+7y3x + 7y cannot be combined. Adding requires a matching index and a matching radicand, and these two share no radicand, so 2+3\sqrt{2} + \sqrt{3} is already in simplest form. Simplify before deciding, though: 12+27\sqrt{12} + \sqrt{27} looks unlike but becomes 23+33=532\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}.Read more →

Why does multiplying by the conjugate remove the radical when squaring doesn't?

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Because they trigger different patterns. A conjugate pair gives a difference of squares with no middle term: (7+3)(7−3)=7−3=4(\sqrt{7} + \sqrt{3})(\sqrt{7} - \sqrt{3}) = 7 - 3 = 4, fully rational. Squaring gives a perfect square trinomial whose middle term still carries a radical: (5+2)2=7+210(\sqrt{5} + \sqrt{2})^2 = 7 + 2\sqrt{10}. The opposite signs are what cancel the cross terms.Read more →

Why is (a+b)2(a + \sqrt{b})^2 not a2+ba^2 + b?

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Because squaring a binomial produces three terms, not two. The expansion is a2+2ab+ba^2 + 2a\sqrt{b} + b, and the middle term 2ab2a\sqrt{b} is the one most often dropped. Concretely (3+2)2=9+62+2=11+62(3 + \sqrt{2})^2 = 9 + 6\sqrt{2} + 2 = 11 + 6\sqrt{2}, not 1111. The same omission produces wrong answers when a radical equation is squared.Read more →

Do you still need to simplify after multiplying radicals?

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Usually yes, because multiplying tends to create perfect power factors that were not there before. 3⋅12=36=6\sqrt{3} \cdot \sqrt{12} = \sqrt{36} = 6 leaves no radical at all, and 6⋅10=60=215\sqrt{6} \cdot \sqrt{10} = \sqrt{60} = 2\sqrt{15} still needs extracting. Division has the mirror problem: the quotient may leave a radical in the denominator to rationalize.Read more →