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Radical Rules



Definition and Basic Rules
Product and Quotient Rules
Power and Exponent Rules
Nested Root Rules
Simplification Rules
Rationalization Rules
Special Value Rules
Domain and Sign Rules
Equation Solving Rules





Definition and Basic Rules

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Definition of nn-th Root
an=b  ⟺  bn=a\sqrt[n]{a} = b \iff b^n = a
The nth root asks what number raised to the nth power gives a
Principal Root Convention
an\sqrt[n]{a} refers to the principal (non-negative) root when n is even
For even roots, we take the positive value by convention
Square Root Rule
a2=∣a∣\sqrt{a^2} = |a|
Square root of a square gives the absolute value
Odd Root Rule
ann=a\sqrt[n]{a^n} = a (when n is odd)
Odd roots preserve the sign of the original number
Even Root Rule
ann=∣a∣\sqrt[n]{a^n} = |a| (when n is even)
Even roots always give non-negative results

Product and Quotient Rules

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Product Rule
an⋅bn=abn\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}
Root of a product equals product of roots
Quotient Rule
anbn=abn\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}
Root of a quotient equals quotient of roots
Product with Different Indices
am⋅bn\sqrt[m]{a} \cdot \sqrt[n]{b} cannot be simplified directly
Different indices require conversion to fractional exponents

Power and Exponent Rules

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Power of a Root
(an)m=amn(\sqrt[n]{a})^m = \sqrt[n]{a^m}
Power of a root can be moved inside the radical
Root of a Power
amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}}
Root of a power becomes fractional exponent
Fractional Exponent Conversion
amn=amn=(an)ma^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m
Fractional exponents and radicals are equivalent
Power Rule for Same Index
(an)n=a(\sqrt[n]{a})^n = a (when defined)
nth power of nth root returns to original value

Nested Root Rules

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Root of a Root
anm=amn=a1mn\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a} = a^{\frac{1}{mn}}
Nested roots multiply their indices
Nested Root Simplification
Multiple nested roots can be combined systematically
Simplify by converting to fractional exponents

Simplification Rules

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Perfect Factor Extraction
an⋅bn=a⋅bn\sqrt[n]{a^n \cdot b} = a \cdot \sqrt[n]{b}
Extract perfect nth powers from under the radical
Like Radical Combination
acn+bcn=(a+b)cna\sqrt[n]{c} + b\sqrt[n]{c} = (a + b)\sqrt[n]{c}
Combine radicals with same index and radicand
Index Reduction
amnmn=amm\sqrt[mn]{a^{mn}} = \sqrt[m]{a^m}
Reduce radical index and exponent by common factors
Radical Factor Cancellation
amnakn=am−kn\frac{\sqrt[n]{a^m}}{\sqrt[n]{a^k}} = \sqrt[n]{a^{m-k}}
Cancel common factors in radical fractions

Rationalization Rules

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Simple Denominator Rationalization
1an=an−1na\frac{1}{\sqrt[n]{a}} = \frac{\sqrt[n]{a^{n-1}}}{a}
Multiply by appropriate radical to clear denominator
Conjugate Rationalization
1a+bc=a−bc(a+bc)(a−bc)\frac{1}{a + b\sqrt{c}} = \frac{a - b\sqrt{c}}{(a + b\sqrt{c})(a - b\sqrt{c})}
Use conjugate to rationalize binomial denominators
Higher Index Rationalization
Complex expressions require systematic approaches
May need multiple steps or different techniques

Special Value Rules

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Root of 1
1n=1\sqrt[n]{1} = 1
Any root of 1 equals 1
Root of 0
0n=0\sqrt[n]{0} = 0
Any root of 0 equals 0
Root of Perfect Powers
bnn=∣b∣\sqrt[n]{b^n} = |b| (even n) or bb (odd n)
Roots of perfect powers simplify directly

Domain and Sign Rules

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Even Root Domain
an\sqrt[n]{a} is defined for a≥0a \geq 0 when n is even (in real numbers)
Even roots of negative numbers are undefined in real numbers
Odd Root Domain
an\sqrt[n]{a} is defined for all real aa when n is odd
Odd roots can handle negative radicands
Sign Preservation
−an=−an\sqrt[n]{-a} = -\sqrt[n]{a} when n is odd
Odd roots preserve and can extract negative signs
Complex Extensions
Even roots of negative numbers exist in complex numbers
−1=i\sqrt{-1} = i in the complex number system

Equation Solving Rules

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Radical Equation Principle
If f(x)n=g(x)\sqrt[n]{f(x)} = g(x), then f(x)=(g(x))nf(x) = (g(x))^n
Raise both sides to the nth power to eliminate radical
Extraneous Solution Check
Always verify solutions in original equation
Raising to even powers can introduce false solutions
Domain Restrictions
Solutions must satisfy original domain constraints
Check that radicands are non-negative for even roots