How to use the exponent rules worksheet generator
- Enter the number of problems, from 1 to 30, and a worksheet number.
- Choose whether negative exponents are included.
- Tick Show the answer key to see the answers, with each result written as a power and as a number.
- Print the sheet for students, and print the key separately for yourself.
Formula
Product rule: a^m x a^n = a^(m+n)
Quotient rule: a^m / a^n = a^(m-n)
Power of a power: (a^m)^n = a^(m x n)
Zero and negative exponents: a^0 = 1, and a^(-n) = 1 / a^n
The rules apply when the bases are the same. A common error is to multiply the bases, so that 5^2 x 5^4 becomes 25^6, when the correct result is 5^6.
The first problems of worksheet number 1
Problem 1 is 5^2 x 5^4. Add the exponents to get 5^6, which is 15625. Problem 2 is 6^9 / 6^3. Subtract the exponents to get 6^6, which is 46656. Problem 3 is (5^3)^3, and multiplying the exponents gives 5^9 = 1953125.
Problem 4 is 6^0 + 6^1, which is 1 + 6 = 7. With negative exponents on, problems such as 3^-2 have an answer of 1/9, and 4^2 x 4^-3 simplifies to 4^-1 = 1/4.
A progression of problems
The sheet rotates through the main rules so that a set of ten problems covers each of them. Start without negative exponents until students are comfortable with products, quotients, powers of powers and the zero exponent. Then turn on negative exponents to bring in reciprocals.
To check a power or a root quickly, use the log, exponent and root calculator. For very large or small numbers, see the scientific notation converter.
Teaching tips
Ask students to expand a small case, such as 5^2 x 5^4 = (5 x 5) x (5 x 5 x 5 x 5), so that the rule is something they see and not only something they remember. Compare a^0 = 1 with the pattern of dividing by the base: 5^2 = 25, 5^1 = 5, 5^0 = 1, and the next step down is 5^-1 = 1/5.
The answers are shown both as a power and as a whole number or a fraction, so students can check either form. The next step after this sheet may be the quadratic equation solver or the factoring quadratics worksheet generator.
Assumptions and limits
- Bases are whole numbers from 2 to 6, and exponents are small whole numbers.
- All problems use a single base, so the rules apply directly.
- With negative exponents on, some problems have fraction answers.
- The sheet is general practice and is not matched to a particular curriculum.
Frequently asked questions
What are the laws of exponents?
The main ones are the product rule, the quotient rule and the power of a power rule, together with a^0 = 1 and a^(-n) = 1 / a^n.
Why is any number to the power 0 equal to 1?
Dividing a^n by a^n gives 1, and by the quotient rule it also gives a^0, so a^0 must be 1 for the rules to agree.
Can I turn off negative exponents?
Yes. The default sheet has only positive exponents and the zero exponent.
How do I get a different sheet?
Change the worksheet number. The same number always makes the same sheet.