Exponent Rules Recap
Before tackling radicals, make sure you have the exponent rules down cold, since radicals are just another way to write fractional exponents.
Product rule: aᵐ · aⁿ = aᵐ⁺ⁿ — when multiplying the same base, add the exponents.
Quotient rule: aᵐ / aⁿ = aᵐ⁻ⁿ — when dividing the same base, subtract the exponents.
Power rule: (aᵐ)ⁿ = aᵐⁿ — when raising a power to a power, multiply the exponents.
Zero exponent: a⁰ = 1 for any nonzero a.
Negative exponent: a⁻ⁿ = 1/aⁿ — a negative exponent means take the reciprocal.
Power of a product: (ab)ⁿ = aⁿbⁿ.
Power of a quotient: (a/b)ⁿ = aⁿ/bⁿ.
These rules apply to all exponents, including fractions and negatives. For instance, x^(1/2) · x^(1/3) = x^(5/6) uses the product rule with fractional exponents. Getting comfortable with these rules is the foundation for everything that follows.
Radical Notation and Rational Exponents
A radical is just a different way to write a fractional exponent.
The key relationship is ⁿ√a = a^(1/n). More generally, ⁿ√(aᵐ) = a^(m/n). So √x = x^(1/2), ³√x = x^(1/3), and ³√(x²) = x^(2/3).
This equivalence is extremely powerful because it lets you use all the exponent rules on radical expressions.
To simplify √x · ³√x, convert to exponents.
x^(1/2) · x^(1/3) = x^(5/6)
⁶√(x⁵)Without rational exponents, this multiplication would be confusing.
About the index
The index of a radical is the small number in the radical sign: for ³√x, the index is 3. For a regular square root √x, the index is 2 (usually not written). When the index is even, the radicand must be non-negative (for real numbers). When the index is odd, the radicand can be any real number.
Understanding this notation deeply allows you to move fluidly between radical form and exponential form, choosing whichever makes a given problem easier.
Simplifying Radicals Step by Step
To simplify a radical means to extract any perfect-power factors from under the radical sign. For square roots, the process is:
- Factor the radicand
Factor the radicand into prime factors or identify perfect square factors.
- Pull out paired factors
Pull out any factor that appears in a pair.
Factor: 72 = 36 × 2 = 6² × 2.
√72 = √(36 · 2)
6√2For cube roots, pull out factors in groups of three.
³√54 = ³√(27 · 2)
3³√2For variables, use the same logic: √(x⁵) = √(x⁴ · x) = x²√x because x⁴ is a perfect square.
Another example: √(50a³b⁴) = √(25 · 2 · a² · a · b⁴)
5ab²√(2a)The key insight is to look for the largest perfect square (or perfect cube, etc.) that divides the radicand. If you miss the largest, you can still simplify further in a second step.
Solver AI will always reduce to the simplest radical form, which is helpful for checking your work.
Rationalizing the Denominator
In standard mathematical writing, we do not leave radicals in the denominator of a fraction. Rationalizing the denominator means rewriting the expression so that the denominator contains no radicals.
For a simple case like 5/√3, multiply both the numerator and denominator by √3.
5/√3 · √3/√3
5√3/3Binomial denominators — use the conjugate
When the denominator is a binomial with a radical, like 3/(2 + √5), multiply by the conjugate.
3(2 − √5)/((2 + √5)(2 − √5)) = 3(2 − √5)/(4 − 5)
= 3(2 − √5)/(−1) = −3(2 − √5)
−6 + 3√5The conjugate works because (a + b)(a − b) = a² − b², which eliminates the radical.
For cube roots in the denominator, you need to multiply by the appropriate expression to make the radicand a perfect cube.
1/³√2 = ³√4/(³√2 · ³√4) = ³√4/³√8
³√4/2While rationalizing may seem like a cosmetic step, it often simplifies further calculations and is expected on exams and in published mathematical work.
Operations with Radicals
You can add, subtract, multiply, and divide radicals, but the rules differ for each operation.
You can only combine like radicals — radicals with the same index and the same radicand. For example, 3√5 + 7√5 = 10√5, but 3√5 + 7√3 cannot be simplified further.
Sometimes simplifying first reveals like radicals: √12 + √27 = 2√3 + 3√3 = 5√3.
Multiply the coefficients together and the radicands together: 2√3 · 5√7 = 10√21. If the indices are the same, √a · √b = √(ab). Then simplify if possible: √6 · √10 = √60 = 2√15.
Similarly, √a / √b = √(a/b). For example, √50/√2 = √25 = 5.
When working with higher-index radicals, the same principles apply: ³√4 · ³√2 = ³√8 = 2.
Mastering these operations requires practice, especially recognizing when to simplify before combining. Work through at least 10-15 problems mixing all four operations to build fluency.
Practice Problems
Test your skills with these problems, then check your answers.
Simplify √(98x⁶y³).
Solution: √(49 · 2 · x⁶ · y² · y)
7x³y√(2y)Simplify x^(2/3) · x^(5/6).
Solution: Add exponents: x^(4/6 + 5/6) = x^(9/6) = x^(3/2)
x√xRationalize 4/(3 − √2).
Solution: Multiply by conjugate: 4(3 + √2)/((3)² − (√2)²) = 4(3 + √2)/(9 − 2)
4(3 + √2)/7Simplify 3√20 − √45 + √80.
Solution: 3(2√5) − 3√5 + 4√5 = 6√5 − 3√5 + 4√5
7√5Rewrite 1/(x^(−3/4)) in radical form.
Solution: x^(3/4)
⁴√(x³)If you want to practice more or see detailed step-by-step solutions, type any radical expression into Solver AI and see the simplification unfold.