SAT Help 24×7
MathChapter 13: Polynomial and Radical Functions

Notes

Radical Expressions Notes

Interpret radical notation, simplify square and cube roots, combine like radicals, and rationalize denominators with exact domain safeguards.

Practice this topic
Reading progress0%
About 30 minutes
On this page

Radicals reverse integer powers. Accurate work depends on the index, the real-number domain, perfect-power factors, and whether simplified radicals are truly alike.

Radical anatomy and nth roots

Anatomy of a radical expression

Text labels identify the index, radical sign, and complete radicand without relying on arrows or color alone.

\[\sqrt[3]{16x}\]
Index
\(3\)

The root being taken; an omitted index means 2.

Radical sign
\(\sqrt{\phantom{16x}}\)

The symbol that denotes a root.

Radicand
\(16x\)

The entire expression beneath the radical bar.

Even and odd indices over the real numbers

Even index

The radicand of a real principal radical must be nonnegative. Thus \(\sqrt{-9}\) is not real.

Odd index

Negative radicands are allowed because odd powers preserve sign: \(\sqrt[3]{-8}=-2\).

Source-supported fractional exponents
\[a^{1/2}=\sqrt a\quad(a\ge0),\qquad a^{1/3}=\sqrt[3]a\]

Product and quotient properties

Radical properties with their required conditions
PropertyRuleCondition
Square-root product\(\sqrt{ab}=\sqrt a\sqrt b\)\(a,b\ge0\)
Square-root quotient\(\sqrt{a/b}=\sqrt a/\sqrt b\)\(a\ge0,b>0\)
Cube-root product\(\sqrt[3]{ab}=\sqrt[3]a\sqrt[3]b\)\(a,b\) real
Cube-root quotient\(\sqrt[3]{a/b}=\sqrt[3]a/\sqrt[3]b\)\(b\ne0\)

Simplify before combining

Worked example

Extract perfect-square factors

Simplify \(\sqrt{108}-\sqrt{12}+2\sqrt3\).

  1. Simplify each radical

    \(\sqrt{108}=\sqrt{36\cdot3}=6\sqrt3\) and \(\sqrt{12}=2\sqrt3\).

  2. Combine like radicals

    \(6\sqrt3-2\sqrt3+2\sqrt3=6\sqrt3\).

\(6\sqrt3\).

Multiplication and rationalization

Conjugates eliminate the radical cross termsA four-stage algebra flow multiplies conjugate binomials and shows the two middle radical terms cancel.
  1. \((a+\sqrt b)(a-\sqrt b)\)

    Only the sign between terms changes.

  2. \(a^2-a\sqrt b+a\sqrt b-b\)

    FOIL produces opposite middle terms.

  3. \(-a\sqrt b+a\sqrt b=0\)

    The radical cross terms are additive inverses.

  4. \(a^2-b\)

    Difference of squares remains.

Worked example

Rationalize a binomial denominator

Rewrite \(\frac{2}{3-\sqrt5}\) with a rational denominator.

  1. Choose the conjugate

    Multiply by \(\frac{3+\sqrt5}{3+\sqrt5}\), which equals \(1\).

  2. Use difference of squares

    The denominator becomes \(3^2-(\sqrt5)^2=9-5=4\).

  3. Simplify

    \(\frac{2(3+\sqrt5)}4=\frac{3+\sqrt5}{2}\).

\(\frac{3+\sqrt5}{2}\).

Common mistakes and traps

  • Assuming \(\sqrt{a+b}=\sqrt a+\sqrt b\).
  • Adding unlike radicals or forgetting to simplify them first.
  • Using \(\sqrt{x^2}=x\) without a nonnegative assumption.
  • Treating a negative even-root radicand as real.
  • Using the same binomial instead of the conjugate during rationalization.
  • Multiplying only one term of a binomial numerator or denominator.
Mini check

Reveal like radicals

Simplify \(\sqrt{75}-\sqrt{27}\).

  1. \(2\sqrt3\)
  2. \(\sqrt{48}\)
  3. \(8\sqrt3\)
  4. \(4\sqrt2\)
Show answer and explanation

Answer: \(2\sqrt3\)

\(\sqrt{75}=5\sqrt3\) and \(\sqrt{27}=3\sqrt3\), so the difference is \(2\sqrt3\).

Key takeaways

Key takeaways

What to remember

  • The index controls which root is taken and which real radicands are allowed.
  • Extract perfect-power factors before combining radicals.
  • Only like radicals combine by addition or subtraction.
  • Use a conjugate to rationalize a binomial radical denominator.
Continue learning

Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.