Exponent rules: 7 key strategies to solve tough equations
All PostsWhat are exponents?
Exponents, also known as powers, are values that show how many times to multiply a base number by itself. For example, 43 is telling you to multiply four by itself three times. 43 = 4 × 4 × 4 = 64The number being raised by a power is known as the base, while the superscript number above it is the exponent or power. [caption id="attachment_4078" align="aligncenter" width="600"]

Exponent rules
There are seven exponent rules, or laws of exponents, that your students need to learn. Each rule shows how to solve different types of math equations and how to add, subtract, multiply and divide exponents.
1. Product of powers rule
When multiplying two bases of the same value, keep the bases the same and then add the exponents together to get the solution.42 × 45 = ?Since the base values are both four, keep them the same and then add the exponents (2 + 5) together.42× 45 = 47Then multiply four by itself seven times to get the answer. 47 = 4 × 4 × 4 × 4 × 4 × 4 × 4 = 16,384Let’s expand the above equation to see how this rule works:
2. Quotient of powers rule
Multiplication and division are opposites of each other -- much the same, the quotient rule acts as the opposite of the product rule.When dividing two bases of the same value, keep the base the same, and then subtract the exponent values.55 ÷ 53 = ?Both bases in this equation are five, which means they stay the same. Then, take the exponents and subtract the divisor from the dividend.55 ÷ 53 = 52Finally, simplify the equation if needed: 52 = 5 × 5 = 25Once again, expanding the equation shows us that this shortcut gives the correct answer:
3. Power of a power rule
This rule shows how to solve equations where a power is being raised by another power.(𝒙3)3 = ?In equations like the one above, multiply the exponents together and keep the base the same.(𝒙3)3 = 𝒙9Take a look at the expanded equation to see how this works:
4. Power of a product rule
When any base is being multiplied by an exponent, distribute the exponent to each part of the base. (𝒙𝑦)3 = ?In this equation, the power of three needs to be distributed to both the 𝒙 and the 𝑦 variables.(𝒙𝑦)3 = 𝒙3𝑦3This rule applies if there are exponents attached to the base as well. (𝒙2𝑦2)3 = 𝒙6𝑦6Expanded, the equation would look like this:
5. Power of a quotient rule
A quotient simply means that you’re dividing two quantities. In this rule, you’re raising a quotient by a power. Like the power of a product rule, the exponent needs to be distributed to all values within the brackets it’s attached to.(𝒙/𝑦)4 = ?Here, raise both variables within the brackets by the power of four.
6. Zero power rule
Any base raised to the power of zero is equal to one.
7. Negative exponent rule
When there is a number being raised by a negative exponent, flip it into a reciprocal to turn the exponent into a positive. Don’t use the negative exponent to turn the base into a negative. [caption id="attachment_4109" align="aligncenter" width="600"]
- Turn the number into a fraction (put it over one)
- Flip the numerator into the denominator and vice versa
- When a negative number switches places in a fraction it becomes a positive number

How Prodigy can help you teach exponents
Prodigy is a curriculum-aligned math game you can use to assign questions, track progress, and identify trouble spots in your students’ learning — for free!With so many different exponent rules to follow and several students to track, it can be hard to see who needs help with what. Prodigy makes it easy to track progress, and create a unique gaming experience for each student based on their needs.
- Prepare students for standardized tests
- Reinforce in-class concepts (like exponent rules!)
- Differentiate math practice in the math classroom and at home


Free exponent rules worksheet
Math worksheets are handy tools that can show how students are understanding key concepts. You can see how students are coming up with answers, where they’re struggling, and if any concepts need to be covered in more detail. [caption id="attachment_4664" align="aligncenter" width="390"]
Conclusion: exponent rules practice
Exponents are used to show how many times a base value is multiplied by itself. This simplifies equations to an easier to read format.(𝒙𝒙𝒙𝒙𝒙𝒙𝒙𝒙𝒙)(𝑦𝑦𝑦𝑦𝑦𝑦)(𝑧𝑧𝑧𝑧𝑧) = 𝒙9𝑦6𝑧5There are seven basic rules that explain how to solve most math equations that involve exponents. The exponent rules are:- Product of powers rule — Add powers together when multiplying like bases
- Quotient of powers rule — Subtract powers when dividing like bases
- Power of powers rule — Multiply powers together when raising a power by another exponent
- Power of a product rule — Distribute power to each base when raising several variables by a power
- Power of quotient rule — Distribute power to all values in a quotient
- Zero power rule — Any base raised to the power of zero becomes one
- Negative exponent rule — To change a negative exponent to a positive one, flip it into a reciprocal
Create or log in to your teacher account on Prodigy -- a free, game-based learning platform that assesses student progress and performance as they play. Aligned with curricula across the English-speaking world, it’s loved by more than a million teachers and 50 million students.
