Key takeaway: Exponent rules are shortcuts for simplifying powers with the same base: add exponents when multiplying, subtract them when dividing, and multiply them when raising a power to a power.
Grade 88.EE.A.1Numbers and Operations
Definition of Exponent Rules
The exponent rules come from what an exponent means, repeated multiplication. Writing \(x^3 \cdot x^4\) in full gives seven copies of \(x\) multiplied together, which is \(x^7\). The rule “add the exponents” is just a shortcut for that counting.
There are three main laws, and each applies only when the bases match. Multiplying powers adds exponents. Dividing powers subtracts them. Raising a power to another power multiplies the exponents. Knowing why each one works prevents the most common slips, such as multiplying exponents when you should add them. Combined with the zero and negative exponent facts, they simplify nearly every expression.
Formula and notation
For a nonzero base \(x\) and integers \(m\) and \(n\):
\[x^m \cdot x^n = x^{m+n}, \qquad \frac{x^m}{x^n} = x^{m-n}, \qquad (x^m)^n = x^{mn}\]
Also \((xy)^n = x^n y^n\).
Key facts
- These exponent rules for products and quotients need the same base on both factors.
- A power of a power multiplies the exponents instead of adding them.
- Each law can be checked by expanding small cases with real numbers.
- A power outside parentheses reaches every factor inside.
Exponent Rules example
Use the exponent rules to simplify \(\dfrac{(x^{4})^{3} \cdot x^{2}}{x^{5}}\).
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Power of a power: \((x^{4})^{3} = x^{12}\).
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Multiply in the numerator: \(x^{12} \cdot x^{2} = x^{14}\).
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Divide: \(\dfrac{x^{14}}{x^{5}} = x^{14-5} = x^{9}\).
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Check with \(x = 2\): \(2^{14} : 2^{5} = 16384 : 32 = 512 = 2^{9}\).
The expression simplifies to \(x^{9}\).
Common mistake
Watch out
Hugo simplifies \(x^{3} \cdot x^{4}\) as \(x^{12}\), multiplying the exponents.
Why it is wrong
Multiplying two powers of one base adds the exponents. Written out, \(x \cdot x \cdot x\) times \(x \cdot x \cdot x \cdot x\) has seven factors, so the answer is \(x^{7}\). Test with \(x = 2\): \(8 \cdot 16 = 128 = 2^{7}\), while \(2^{12} = 4096\).
Related terms and lessons
- Grade 8 · Properties of Integer ExponentsLessonWorksheetQuiz
- Algebra 1 · Exponent Rules, Radicals and Rational ExponentsLessonWorksheetQuiz
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