Pre-Algebra
Numbers and operations
Integer exponents and exponent rules
20 practice questions
0 video lessons
Theory + worked examples
Integer Exponents and Exponent Rules
Texas Pre-Algebra (TEKS) • Standard 8.2(B) • Numbers & Operations
Integer Exponents and Exponent Rules is a topic in Numbers & Operations in the Texas Essential Knowledge and Skills. It is aligned to Standard 8.2(B), which requires students to apply the properties of integer exponents to generate equivalent expressions.
Exponents show repeated multiplication; the exponent rules add, subtract, or multiply exponents to combine powers.
Theory
An exponent counts repeated multiplication: \(a^n\) means \(a\) multiplied \(n\) times.
- Product: \(a^m\cdot a^n=a^{m+n}\).
- Quotient: \(a^m\div a^n=a^{m-n}\).
- Power: \((a^m)^n=a^{mn}\).
- Zero & negative: \(a^0=1\), \(a^{-n}=\dfrac{1}{a^n}\).
Add exponents to multiply, subtract to divide.
The exponent rules.
What an exponent means.
Key rules:
\[a^m a^n=a^{m+n},\quad (a^m)^n=a^{mn},\quad a^{-n}=\dfrac{1}{a^n}\]
\(a^0=1\) for any nonzero \(a\).
How to apply exponent rules
- Same base, multiplying: add exponents.
- Same base, dividing: subtract exponents.
- Power of a power: multiply exponents.
- Negative exponent: take the reciprocal.
Example 1 — Product rule
Simplify \(3^2\cdot 3^4\).
Solution
Add exponents.
| \(3^{2+4}\) | \(=\) | \(3^6\) |
Example 2 — Quotient rule
Simplify \(\dfrac{5^7}{5^3}\).
Solution
Subtract exponents.
| \(5^{7-3}\) | \(=\) | \(5^4\) |
Example 3 — Power rule
Simplify \((2^3)^2\).
Solution
Multiply exponents.
| \(2^{3\cdot2}\) | \(=\) | \(2^6\) |
Example 4 — Negative exponent
Write \(4^{-2}\) as a fraction.
Solution
A negative exponent is a reciprocal.
| \(4^{-2}\) | \(=\) | \(\dfrac{1}{4^2}=\dfrac{1}{16}\) |
Common pitfalls
\(3^2\cdot3^4=3^6\), not \(9^6\) — the base stays.
\(a^{-n}\) is a reciprocal, not a negative number.
\(a^0=1\), not \(0\).
Frequently asked questions
How do I multiply powers with the same base?
Add the exponents.
What does a negative exponent mean?
The reciprocal: \(a^{-n}=1/a^n\).
What is \(a^0\)?
\(1\) for any nonzero \(a\).
What is \((2^3)^2\)?
\(2^6=64\).
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