Key takeaway: Rational exponents are fraction exponents in which the denominator tells the root and the numerator tells the power, so the fraction packs a root and a power into a single small number written above the base.
Grade 9 – Algebra 1N-RN.A.1Numbers and Operations
Definition of Rational Exponents
Rational exponents extend the idea of an exponent from whole numbers to fractions. An exponent of the form \(\frac{1}{n}\) asks for the \(n\)th root, which means \(625^{1/4}\) equals 5, since multiplying four fives gives 625. A fraction with a bigger numerator mixes a root with a power inside one tiny symbol.
Read the denominator first for the root, then use the numerator as the power. The order is flexible: you can take the root first and then the power, or the other way. Taking the root first usually gives smaller numbers, which makes the work easier by hand. Negative rational exponents also flip the number to its reciprocal.
Formula and notation
Take a positive base \(a\) and whole numbers \(m\) and \(n\) with \(n > 0\). Root-then-power and power-then-root give the same value:
\[a^{m/n} = \sqrt[n]{a^{m}}\]
A minus sign in front of the fraction flips the result: \(a^{-m/n}\) is \(1\) divided by \(a^{m/n}\). Every familiar exponent law keeps working with fractions in the exponent.
Key facts
- Bottom of the fraction: which root to take.
- Half as an exponent means square root, and one third means cube root.
- A minus sign in the exponent makes a reciprocal; the fraction still picks the root.
- Take the root first when the base is a perfect power, to keep numbers small.
Rational Exponents example
Evaluate \(128^{3/7}\) and \(49^{-3/2}\).
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Seventh root first: \(\sqrt[7]{128} = 2\) because \(2^{7} = 128\). Then cube it: \(2^{3} = 8\).
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For the second one, \(\sqrt{49} = 7\), and \(7^{3} = 343\), so \(49^{3/2} = 343\).
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The minus sign in the exponent means a reciprocal: \(49^{-3/2} = \frac{1}{343}\).
\(128^{3/7} = \) 8 and \(49^{-3/2} = \) \(\frac{1}{343}\).
Common mistake
Watch out
Pablo reads \(128^{3/7}\) as \(\frac{128^{3}}{7}\) and gets about \(299{,}593.1\).
Why it is wrong
The exponent \(\frac{3}{7}\) is a single number, not a power followed by a division. The 7 says which root to take and the 3 says which power to use: the seventh root of 128 is 2, and \(2^{3} = 8\). An answer near three hundred thousand for such a small base should look suspicious.
Related terms and lessons
- Algebra 1 · Exponent Rules, Radicals and Rational ExponentsLessonWorksheetQuiz
- Algebra 2 · Radicals and Rational ExponentsLessonWorksheetQuiz
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