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Algebra 2 Radical functions

Rational exponents

20 practice questions 0 video lessons Theory + worked examples

Rational Exponents

California Algebra 2 • Standard N-RN.1 • Radical Functions

Rational Exponents is a topic in Radical Functions in the California Common Core State Standards. It is aligned to Standard N-RN.1, which requires students to explain and use the definition of rational exponents in terms of radicals.

A rational exponent combines a power and a root: \(x^{m/n}=\sqrt[n]{x^m}\), and the usual exponent rules still apply.

California Algebra 2 › Radical Functions › Rational Exponents  —  Standard N-RN.1

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Theory

A rational exponent is a fraction: the denominator is a root, the numerator a power.

\[x^{m/n}=\sqrt[n]{x^{m}}=\left(\sqrt[n]{x}\right)^{m}.\]

All the usual exponent rules (product, power, negative) still hold.

Denominator = root, numerator = power — take the root first to keep numbers small.
Rational exponents Rational exponents Rational exponents x^(1/n) = ⁿ√x x^(m/n) = ⁿ√(x^m) = (ⁿ√x)^m numerator: power, denominator: root
Rational exponents as roots and powers.
Exponent rules still apply Exponent rules still apply Exponent rules still apply xᵃ · xᵇ = xᵃ⁺ᵇ (xᵃ)ᵇ = xᵃᵇ x⁻ᵃ = 1 / xᵃ
The exponent rules carry over.

The definition:

\[x^{1/n}=\sqrt[n]{x},\qquad x^{m/n}=\left(\sqrt[n]{x}\right)^{m}\]
x to the one over n is the n-th root; x to the m over n is that root to the m
Negative exponent gives a reciprocal, \(x^{-a}=\dfrac1{x^a}\).

How to evaluate

  1. Read the denominator as the root.
  2. Read the numerator as the power.
  3. Take the root first, then apply the power.
  4. For a negative exponent, take the reciprocal.
Example 1 — Simple root
Evaluate \(8^{1/3}\).
Solution

The denominator \(3\) is a cube root.

\(8^{1/3}\)\(=\)\(\sqrt[3]{8}=2\)
8 to the one third is 2
Example 2 — Power and root
Evaluate \(16^{3/4}\).
Solution

Take the fourth root, then cube.

\(16^{3/4}\)\(=\)\((\sqrt[4]{16})^3\)
\(=\)\(2^3=8\)
16 to the three fourths is 8
Example 3 — Radical to exponent
Write \(\sqrt[3]{x^2}\) with a rational exponent.
Solution

Root is the denominator, power the numerator.

\(\sqrt[3]{x^2}\)\(=\)\(x^{2/3}\)
the cube root of x squared is x to the two thirds
Example 4 — Negative rational exponent
Evaluate \(27^{-2/3}\).
Solution

Negative exponent means reciprocal.

\(27^{-2/3}\)\(=\)\(\dfrac{1}{27^{2/3}}\)
\(=\)\(\dfrac{1}{9}\)
27 to the negative two thirds is one ninth

Common pitfalls

Denominator is the root, not the power.
Take the root first to avoid large numbers.
Negative exponents mean reciprocal, not a negative value.

Frequently asked questions

What does \(x^{1/2}\) mean?

The square root of \(x\).

What does \(x^{m/n}\) mean?

The \(n\)-th root of \(x^m\).

How do you evaluate \(8^{2/3}\)?

Take the cube root (2), then square: \(4\).

Do exponent rules apply to rational exponents?

Yes — product, power, and negative-exponent rules all hold.

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