Math PicnicEvery sheet is a promise

Negative rational exponents worksheets

An exponent such as −4/3 has three instructions in it: a root, a power and a flip. These free printable worksheets give ten of them a page, always ending in a fraction with 1 on top, and the answer key works through all three parts one line at a time.

Customize Save as PDF I want another one

If they get stuck

See it worked step by step →
Negative rational exponents
Negative rational exponents worksheet — Answers are unit fractions, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

Take 8^(−4/3). The 3 on the bottom says to take the cube root, which gives 2. The 4 on top says to raise that to the 4th power, which gives 16. The minus sign says to put it under 1, which gives 1/16. Keeping the three parts separate is what makes this manageable, and the key sets them out in that order on every question.

Why this sheet works

The mistake everyone makes first is reading the minus sign as a negative answer. 8^(−4/3) is not −16. It is a small positive number, and all the answers on this page are positive fractions. A negative exponent is about where the number sits in a fraction, not about which side of zero it falls on.

How this one works

25^(−3/2)

√25 = 5
5³ = 125
125→
25^(−3/2)=
  1. The BOTTOM of the exponent says which root to take, so the square root of 25 comes first. Root before power, always — the other order works and makes the numbers enormous.
  2. The TOP says what to raise it to. 5 to the power 3 is 125.
  3. Last, the MINUS. A negative exponent does not make the answer negative — it turns it upside down. So 25^(3/2) is 125, and 25^(−3/2) is one over that.
  4. So 25^(−3/2) is .

Pick another skill