In “Explore” two patterns were shown: 1) when expressions with the same base are multiplied, the exponents are added, and 2) when raising a base with a power to a power, the exponents are multiplied.
We can generalize these into two rules of exponents with proofs as follows.
Rule 1: Multiplication
- When you multiply values with the same base, you add the exponents.
- Proof:
\(a^n\cdot a^m=\)
\((a\cdot a\cdot a\cdot…) \ _{n \ times} \cdot (a\cdot a\cdot a\cdot…)\ _{m \ times}\) which is the same as \((n+m) \ a\)’s multiplied together.
Therefore: \(a^n\cdot a^m=a^{n+m}\)
Rule 2: Raise a Power to a Power
- When you raise a base with a power to a power, you multiply the exponents.
- Proof:
\((a^n)^m=\)
\((a^n\cdot a^n\cdot a^n\cdot …) \ _{m \ times}\) which is the same as \(a^n \ m\) times.
Therefore: \((a^n)^m=a^{n\cdot m}\)
Simplify the following, using the two exponent rules above:
1. \(3x^2\cdot 5x^3\)
2. \(4x^2y^3\cdot8xy^2\)
3. \(2x^{-1}\cdot3x^3\)
4. \((2xy^3)(5x^3y)\)
5. \((2x^2y^3)^2\)
6. \((5^2)^{-1}\)
7. \((x^{-1}y^{-1})^{-2}\)
