Exponents • Topic 3 of 4

Rational Exponents

A fractional (rational) exponent combines a power and a root: x^(m/n) = ⁿ√(xᵐ) = (ⁿ√x)ᵐ. The denominator is the root and the numerator is the power. So x^(1/2) = √x, x^(1/3) = ∛x, and 8^(2/3) = (∛8)² = 4. It is usually easiest to take the root first (smaller numbers) then apply the power. All exponent laws extend to rational exponents. Converting between radical and rational-exponent notation is a frequent SAT task, especially when simplifying or comparing expressions with roots.

✅ Solved examples

1. Write √x using a rational exponent.
A square root is the one-half power: x^(1/2).
2. Evaluate 9^(1/2).
9^(1/2) = √9 = 3.
3. Evaluate 8^(2/3).
Take the cube root then square: (∛8)² = 2² = 4.
4. Write ∛(x²) using a rational exponent.
(x²)^(1/3) = x^(2/3).

✏️ Practice — try these, take hints as needed

1. Evaluate 16^(1/2).
Half power = square root.
√16.
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4.
2. Evaluate 27^(1/3).
Third power = cube root.
∛27.
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3.
3. Evaluate 16^(3/4).
Fourth root first: ⁴√16 = 2.
Then cube: 2³.
—
8.
4. Write ⁴√x using a rational exponent.
Denominator is the root.
Root 4 → 1/4.
—
x^(1/4).
5. Evaluate 32^(2/5).
Fifth root: 32^(1/5) = 2.
Then square.
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4.

📝 Topic test — 8 questions

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