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Rational Exponents

A fractional exponent is not a new operation — it is a root written in exponent form, so that every law from the previous section keeps working. The denominator says which root, the numerator says which power. Once that reading is automatic, the only two questions left are which to do first (the root) and whether the answer exists at all (check the denominator for even, and the base for negative).

Class Notes — Chapter 1, Section 1.2

Calculator
The definitions
Two definitions carry the whole section; everything after them is the old laws again.
  • Unit fraction is a root

    The denominator is the index of the root. This is a definition, not a consequence — it is chosen so the power-of-a-power rule survives.

  • Why that definition

    Something that gives back when raised to the is exactly what the th root means. The definition is forced.

  • General rational exponent

    Both readings give the same number. Take the root first — it keeps the numbers small enough to do in your head.

  • Negative rational exponent

    The minus still just means move across the bar. It does not flip the fraction inside the exponent.

  • Negative fraction base

    Flip the base, then work with a positive exponent.

  • All the old laws still hold

    Product, quotient, power of a power, power of a product — unchanged. The only new work is adding and multiplying fractions.

Watch the bracket, exactly as in the integer case: because the exponent binds only to the 64, while is not a real number at all.

Does the answer exist?
Check this before doing any work. It costs three seconds and it is the difference between an answer and a wrong answer.

  1. Base positive — always fine. Any rational exponent on a positive base gives a positive real number. No check needed.

  2. Base negative, denominator odd — fine. An odd root of a negative number is a negative number: , because .

  3. Base negative, denominator even — not a real number. has no real value: no real number to the fourth power is negative. Say so rather than guessing a sign.

  4. Base zero, exponent negative — undefined. would be . Division by zero, so it is undefined.

  5. Reduce the fraction before deciding. looks even-rooted, but , and the cube root is fine. Always reduce first.

Worked examples — evaluating
These want a single number with no exponents left. The move that makes them easy is always the same: root first, power second.

Worked examples — simplifying
These want letters, not numbers, and the answer must carry only positive exponents. It is the same four-step clean-up as integer exponents, with fraction arithmetic on the exponents.

True or false
Both of these come from reading the minus sign as though it lived inside the fraction.

  • Negating an exponent is not the same as turning it upside down. A minus moves the power across the bar and leaves exactly as it was. Test at : , while .

  • The reciprocal belongs to the whole power, not to the exponent. is something else entirely — it is . Test at : , while .

  • , not . An even root of a negative base has no real value, so the right answer is to say so — not to attach a minus to the positive root.

Practice
Root first, then power, then tidy the sign.

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