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Algebra 2 Unit 5: Radicals & rational exponents

Radicals and rational exponents are two ways to write roots. Converting between them makes simplifying easier.

Covers: Simplifying radicals, rational exponents, operations with radicals, solving radical equations

Study guide

A radical expresses a root: √x is the square root, ∛x the cube root, and ⁿ√x the nth root, where n is the index. Even roots of negative numbers aren't real, but odd roots are (∛−8 = −2). To simplify a square root, pull out perfect-square factors: √18 = √9·√2 = 3√2. For cube roots, pull out perfect cubes.

Rational exponents are another way to write radicals: x^(1/n) = ⁿ√x and x^(m/n) = (ⁿ√x)ᵐ. All the exponent rules still apply, so 8^(2/3) = (∛8)² = 4. Converting between forms often makes a problem easier.

Operations with radicals: you can only add or subtract like radicals, with the same index and radicand, so √2 + √8 = √2 + 2√2 = 3√2. Multiply radicals with the same index by multiplying radicands: √3·√12 = √36 = 6. To rationalize a denominator, multiply by a radical that makes it a perfect power; for binomial denominators like 1 + √2, multiply by the conjugate 1 − √2.

To solve a radical equation, isolate the radical, raise both sides to the index's power (square for square roots), and solve. Squaring can create extraneous solutions, so check every answer in the original equation.

Isolate the power, then raise both sides to the reciprocal exponent n/m: if x^(3/2) = 27, then x = 27^(2/3) = 9. If m is even, consider both ± roots and check them in the original equation. Write each radical as a power, such as ⁿ√(xᵃ) = x^(a/n), apply the exponent rules, and then convert back if the answer must be in radical form.

Key ideas

• a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)^m.
• Simplify radicals by pulling out perfect powers.
• Add radicals only when the radicands match.
• Solve radical equations by isolating the radical and raising both sides to a power.
• Always check for extraneous solutions.

Formulas and rules

Common mistakes

• Adding unlike radicals, like √2 + √3 = √5.
• Not checking solutions after squaring.

Words to know

Rational exponent

x^(m/n) = (ⁿ√x)ᵐ, so x^(1/3) = ∛x.

Index of a radical

The small n in ⁿ√x: which root you take.

Simplifying radicals

Pull out perfect-square factors: √18 = 3√2.

Like radicals

Radicals with the same radicand; only these combine (√2 + 2√2 = 3√2).

Squaring both sides

Can introduce extraneous solutions, so always check.

nth root

ⁿ√a: a number whose nth power is a.

Radicand

The number under the radical sign.

Odd root of a negative

Real: ∛−8 = −2.

Multiplying radicals

√a·√b = √(ab) for the same index.

Rationalizing with a conjugate

Multiply by a − √b to clear a + √b from a denominator.

Radical equation

An equation with the variable under a radical; isolate, then raise to the index.

Sample practice questions

35 more questions like these, from level 1 to 5, are in the practice set.