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Algebra 2 Radical functions

Rational exponents

20 practice questions 0 video lessons Theory + worked examples

Rational Exponents

Texas Algebra II (TEKS) • Standard 2A.7(G) • Radical Functions

Rational Exponents is a topic in Radical Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.7(G), which requires students to rewrite radical expressions using rational exponents and vice versa.

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

Texas Algebra II (TEKS) › Radical Functions › Rational Exponents  —  Standard 2A.7(G)

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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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