Exponents and radicals
Multiplying and dividing powers worksheets
Choose multiplication, division or a balanced mix for 12 exponent-law problems. Use positive or mixed signed exponents with exact simplified answers.
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Simplifying rational exponents worksheets
Simplify ten exact monomial powers, products and quotients with rational exponents. Choose easy, medium or hard profiles and print a worked answer key.
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Assume every variable is positive. Combine matching variable powers, reduce every exponent and write only positive exponents with no fractional exponent in a denominator.
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Combine exact rational exponents on positive variables
Every variable is positive. This stated domain makes the product, quotient and nested-power laws used here valid without claiming a rule for every real or complex input.
The key shows exact coefficient and exponent arithmetic, then rewrites negative totals with a positive fractional power above the bar and an integer power below it.
The preview and downloaded PDF use the same resolved questions. Answers follow on separate pages; choose the student-only edition when handing out the exercise.
Learning objective
What this activity builds
Combine rational exponents exactly and write an equivalent form with positive exponents and integer powers in every denominator.
Worked example
Simplify x to the power negative three-halves.
x to the power one-half over x squared.
One-half minus two is negative three-halves. The denominator exponent is the integer 2.
Teaching notes
Make the practice count
- Handle coefficients separately from variable exponents.
- Add exponents in products and subtract them in quotients.
- Multiply inner and outer exponents for nested powers.
- Use the positive-variable assumption when applying rational power laws.
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Customize the practice
Choose your options, update the preview, then download the exact same pages.
Choose one or more profiles for ten questions. Every variable is positive, so the displayed rational-power laws are valid. Answers use positive exponents and integer powers in denominators.
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