Exponent Rules
Exponent rules are shortcuts for simplifying powers. When the bases match, multiply by adding the exponents: x^a \cdot x^b = x^{a+b}. Divide by subtracting them. For a power of a power, multiply the exponents. For a power of a product or quotient, give the exponent to each factor.
Before this lesson: You should know what exponents mean, how to work with algebraic expressions, and how to do combining like terms.
Why the Exponent Rules Work
An exponent counts how many times a base is multiplied. So x^3 means x \cdot x \cdot x. The base is x and the exponent is 3.
Now multiply x^3 \cdot x^2. Write out every factor: (x \cdot x \cdot x)(x \cdot x). That is 5 factors of x, so the answer is x^5. You added 3 and 2.
Raise x^2 to the 3rd power and you get x^2 \cdot x^2 \cdot x^2. That is 3 groups of 2, or 6 factors. So (x^2)^3 = x^6. You multiplied 2 and 3.
The Exponent Rules
In each rule, a and b are positive whole numbers. The bases can’t be 0 when they are on the bottom of a fraction.
| Rule | In symbols | Example |
|---|---|---|
| Product rule | x^a \cdot x^b = x^{a+b} | x^4 \cdot x^3 = x^7 |
| Quotient rule | \dfrac{x^a}{x^b} = x^{a-b} | \dfrac{x^9}{x^4} = x^5 |
| Power of a power | (x^a)^b = x^{ab} | (x^5)^2 = x^{10} |
| Power of a product | (xy)^a = x^a y^a | (3x)^2 = 9x^2 |
| Power of a quotient | \left(\dfrac{x}{y}\right)^a = \dfrac{x^a}{y^a} | \left(\dfrac{x}{2}\right)^3 = \dfrac{x^3}{8} |
How to Simplify with Exponent Rules
- Multiply or divide the numbers (the coefficients) first.
- Group the same bases together.
- Use the rules on each base: add exponents to multiply, subtract to divide, multiply for a power of a power.
- If a whole product is raised to a power, raise every factor, including the number.
- Write each base once, in alphabetical order.
In this lesson, the quotient rule leaves a positive exponent. What x^0 and negative exponents mean is the next lesson.
Worked Examples
Example 1: Product Rule
Simplify (a) 2^3 \cdot 2^4 and (b) x^5 \cdot x \cdot x^2.
(a) The bases match, so add the exponents: 2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128. Keep the base 2. Do not multiply the bases.
(b) A plain x has exponent 1. So x^5 \cdot x^1 \cdot x^2 = x^{5+1+2} = x^8.
Answer: (a) 2^7 = 128; (b) x^8
Example 2: Monomials with Coefficients
Simplify (3x^4y)(-5x^2y^3).
Step 1: Multiply the numbers: 3 \cdot (-5) = -15.
Step 2: Add the exponents on each base: x^{4+2} = x^6 and y^{1+3} = y^4.
Answer: -15x^6y^4
Example 3: Quotient Rule
Simplify \dfrac{24a^7b^3}{6a^2b}.
Step 1: Divide the numbers: 24 \div 6 = 4.
Step 2: Subtract the exponents on each base: a^{7-2} = a^5 and b^{3-1} = b^2.
Answer: 4a^5b^2
Example 4: Power Rules
Simplify (a) (2x^3)^4, (b) (-3y^2)^3 and (c) \left(\dfrac{x^2}{5}\right)^3.
(a) Raise each factor to the 4th power: 2^4 \cdot (x^3)^4 = 16x^{12}.
(b) Raise each factor to the 3rd power: (-3)^3 \cdot (y^2)^3 = -27y^6. An odd power of a negative number is negative.
(c) Raise the top and the bottom: \dfrac{(x^2)^3}{5^3} = \dfrac{x^6}{125}.
Answer: (a) 16x^{12}; (b) -27y^6; (c) \dfrac{x^6}{125}
Example 5: Word Problem
A cube has edges that are 3x^2 centimeters long. Write its volume as a single term. Then find the volume when x = 2.
Step 1: Volume of a cube = edge cubed. So V = (3x^2)^3.
Step 2: Use the power of a product rule: 3^3 \cdot (x^2)^3 = 27x^6.
Step 3: Put in x = 2: 27 \cdot 2^6 = 27 \cdot 64 = 1728. Check: the edge is 3 \cdot 2^2 = 12 cm, and 12^3 = 1728.
Answer: The volume is 27x^6 cubic centimeters. When x = 2, it is 1,728 cubic centimeters.
Common Mistakes
| Mistake | Example of the error | Fix |
|---|---|---|
| Multiplying exponents when multiplying powers | x^3 \cdot x^4 = x^{12} | Add them: x^3 \cdot x^4 = x^7 |
| Multiplying the bases | 2^3 \cdot 2^4 = 4^7 | Keep the base: 2^3 \cdot 2^4 = 2^7 |
| Combining different bases | x^2 \cdot y^3 = (xy)^5 | x^2y^3 can’t be simplified |
| Not raising the number | (2x^3)^4 = 2x^{12} | Raise every factor: 16x^{12} |
| Adding exponents when adding terms | x^2 + x^3 = x^5 | These are not like terms. x^2 + x^3 stays as it is |
| Using the product rule on a sum | (x + y)^2 = x^2 + y^2 | The rule is only for products: (xy)^2 = x^2y^2 |
Checking Your Work with Numbers
Not sure a step is right? Put in a small number and compare both sides. Try x = 2.
| Step to test | Left side | Right side | Match? |
|---|---|---|---|
| x^3 \cdot x^4 = x^7 | 8 \cdot 16 = 128 | 2^7 = 128 | Yes |
| x^3 \cdot x^4 = x^{12} | 128 | 2^{12} = 4096 | No, so this step is wrong |
| (x^2)^3 = x^6 | 4^3 = 64 | 2^6 = 64 | Yes |
Exponent rules are used all the time with powers of 10. In scientific notation, (3 \times 10^4)(2 \times 10^5) = 6 \times 10^9, because 10^4 \cdot 10^5 = 10^9. You will also use them to multiply polynomials in Unit 7.
Practice Problems
- Simplify x^6 \cdot x^3.
- Simplify (4a^2)(7a^5).
- Simplify \dfrac{y^{10}}{y^4}.
- Simplify (m^3)^5.
- Simplify (-2x^4)^3.
- Simplify \dfrac{18x^8y^5}{3x^3y^2}.
- Simplify \left(\dfrac{a}{3}\right)^4.
- Simplify (2x^2y)^3(xy^2).
- Find the value of \dfrac{5^9}{5^7} without a calculator.
- A square tile has sides 4n^3 inches long. Write its area as a single term.
Show answers
- x^{6+3} = x^9
- 4 \cdot 7 = 28 and a^{2+5} = a^7, so 28a^7.
- y^{10-4} = y^6
- m^{3 \cdot 5} = m^{15}
- (-2)^3 \cdot x^{12} = -8x^{12}
- 18 \div 3 = 6, x^{8-3} = x^5, y^{5-2} = y^3. So 6x^5y^3.
- \dfrac{a^4}{3^4} = \dfrac{a^4}{81}
- (2x^2y)^3 = 8x^6y^3. Then 8x^6y^3 \cdot xy^2 = 8x^7y^5.
- 5^{9-7} = 5^2 = 25
- A = (4n^3)^2 = 16n^6 square inches.
FAQ
What are the exponent rules?
The main rules are the product rule (x^a \cdot x^b = x^{a+b}), the quotient rule (x^a \div x^b = x^{a-b}) and the power rules. The power rules say (x^a)^b = x^{ab} and (xy)^a = x^a y^a.
When do you add exponents and when do you multiply them?
Add them when you multiply two powers with the same base, like x^2 \cdot x^3 = x^5. Multiply them when you raise a power to a power, like (x^2)^3 = x^6.
Can you use exponent rules with different bases?
Not the product and quotient rules. x^2 \cdot y^3 stays as it is. But if the exponents match, you can group them: x^3y^3 = (xy)^3.
Is (x + y)² the same as x² + y²?
No. The power rule works for products, not sums. For example, (2 + 3)^2 = 25, but 2^2 + 3^2 = 13.
Summary
- Product rule: same base, add the exponents. x^a \cdot x^b = x^{a+b}
- Quotient rule: same base, subtract the exponents. x^a \div x^b = x^{a-b}
- Power of a power: multiply the exponents. (x^a)^b = x^{ab}
- Power of a product or quotient: raise every factor, including numbers.
- Numbers multiply or divide as usual. Different bases can’t be combined.
Previous lesson: Systems of Inequalities
Next lesson: Zero and Negative Exponents (coming soon)
Back to the Algebra 1 Index