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Operations on Roots



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



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.



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.

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.

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.

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.