Solve (a⁵)⁷: Simplifying Power of Power Expressions

When you have $(a^5)^7$, you're multiplying $a^5$ by itself 7 times. This gives you $a^5 \cdot a^5 \cdot a^5 \cdot a^5 \cdot a^5 \cdot a^5 \cdot a^5$, which equals $a^{35}$ using the product rule!

Great question! $a^5 \cdot a^7 = a^{5+7} = a^{12}$ (you add exponents when multiplying powers). But $(a^5)^7 = a^{5 \times 7} = a^{35}$ (you multiply exponents when raising a power to a power).

Apply the power to everything inside the parentheses: $(3a^5)^7 = 3^7 \cdot (a^5)^7 = 2187a^{35}$. Don't forget to raise the coefficient to the power too!

Absolutely! The rule works the same way: $(a^{-3})^4 = a^{-3 \times 4} = a^{-12}$. Just multiply the exponents as usual, keeping track of negative signs.

• Multiply powers: $a^m \cdot a^n = a^{m+n}$ (add exponents)
• Power of power: $(a^m)^n = a^{m \times n}$ (multiply exponents)

Tip: Look for parentheses around the base with an exponent!