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Radicals and Rational Exponents

SAT MathAdvanced MathMedium

Translate between radicals and fractional exponents and simplify perfect-power factors.

What you need to know

  • x¹⁄ⁿ is the nth root of x.
  • xᵐ⁄ⁿ is the nth root of x raised to m.
  • Only like radicals combine by addition.
  • For real even roots, radicands must be nonnegative.

The AYLOTI method

  1. Factor out the largest perfect square or cube.
  2. Rewrite using exponent form when useful.
  3. Combine only identical radical parts.
  4. Check domain restrictions.

AYLOTI • Sir Hunain Zia • SAT Score 1594/1600 • 154 Total A Grades (Personal), 11 World Records, 7 Distinctions

Worked example

Simplify √72.

  1. 72 = 36 × 2.
  2. √72 = √36 · √2.
  3. √36 = 6.

Answer: 6√2.

Common SAT traps

  • Claiming √(a + b) = √a + √b.
  • Combining unlike radicals.
  • Ignoring domain restrictions after squaring.
Quick check

Rewrite x³⁄² in radical form.

x√x for x ≥ 0.

Before you move on

  • Explain the core rule without looking back.
  • Redo the worked example using your own steps.
  • Write the trap you are most likely to fall for.
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