How is a problem like 8^(2/3) meant to be solved?

Root first, then power: take the cube root of 8 to get 2, then square it to get 4. Every problem in this practice set is built from a perfect power specifically so the root comes out clean.

Why do some bases in this practice set look unusual, like 256 or 625?

Every base is deliberately a perfect nth power — 625 is 5^4, for instance — so its root is always a whole number instead of an ugly decimal, keeping the focus on the exponent rules rather than root-finding by hand.

Can the base ever be negative?

Yes, at Hard difficulty, and only with an odd root index, like (−32)^(1/5). An even root of a negative base has no real value, so this practice set never produces one.

Why do some answers come out as fractions?

A negative rational exponent, like 16^(-3/4), still means "take the reciprocal" exactly as it does for integer exponents — so the correct answer is a fraction such as 1/8, not a decimal error.

Do I need to know Integer Exponents before this practice set?

Yes. Every rule here — product, quotient, power, negative exponent — is the same one from Integer Exponents, just with a fraction in the exponent instead of a whole number. That lesson is the right starting point if any rule feels unfamiliar.

Can I regenerate as many times as I like?

Not a single one. Generate unlimited sets, completely free, with zero sign-up needed.

Rational Exponents Practice Problems

A fractional exponent packs a root and a power into one symbol, and the Rational Exponents lesson covers exactly how that works. This practice set is where that understanding turns into speed: unlimited fresh problems, always built from clean perfect powers, with instant answer checking.

How the difficulty levels work

Difficulty	What it drills	Example
Easy	A single clean root,
Medium	Root then power,
Hard	Negative bases, negative exponents

Easy is a single root: where is always a perfect th power, so the answer is a clean whole number every time.

Medium introduces the full pattern — take the root first, then raise it to the power — plus, occasionally, a negative rational exponent that resolves to a fraction.

Hard covers the two subtleties from the lesson’s Domain section: odd roots of negative bases (which are real and negative, like ), and negative rational exponents on larger perfect powers.

Worked examples

One example at each difficulty level, solved the way the practice set expects.

Easy. Evaluate .

The denominator is the root: .

Medium. Evaluate .

Take the root first to keep the numbers small: , then cube: .

Hard. Evaluate and .

An odd root of a negative number is negative: , so . For , the negative sign means a reciprocal: .

Using the practice problems

Set the difficulty level and count you want, and a new set of problems appears immediately. Enter whole-number answers as-is, and fraction answers either as a fraction (1/8) or a decimal (0.125) — both are accepted.

Check Answers tallies correct and incorrect answers and highlights the trouble spots
Show Answers reveals the whole answer key instantly, useful for reviewing printed work
Print Worksheet builds a clean, printable page suited for classroom handouts or homework

A reliable strategy for every problem here: take the root first, then apply the power. Rooting a large number first (rather than raising it to a power first, which quickly produces enormous intermediate numbers) is exactly the order the lesson recommends.

The rules these practice problems drill

Rule	Statement
Definition
Full rational exponent
Negative rational exponent
Even root of a negative base	Not a real number
Odd root of a negative base	Defined, and negative

See the Rational Exponents lesson for the full derivation of each rule and worked examples.

Common mistakes to watch for

Powering first instead of rooting first. computed as , then , is correct but much harder than rooting first: .

Treating a negative rational exponent as a negative result. , a positive fraction — the minus sign still just means “reciprocal.”

Assuming every root of a negative number is undefined. Only even roots are. is an ordinary real number, since 5 is odd.

Mixing up which part of the fraction is the root. In , the denominator is the root, not the numerator.

Where to go next

If any rule here feels unfamiliar, Rational Exponents derives every one of them from scratch, and its prerequisite, Integer Exponents, covers the whole-number version of the same rules. For more whole-number exponent drills, see the Integer Exponent Practice Problems.

Rational Exponents Practice Problems