Solve (7^8)^9: Understanding Nested Power Expressions

The power of a power rule says $(a^m)^n = a^{m \times n}$. Think of it this way: $(7^8)^9$ means you're multiplying $7^8$ by itself 9 times, which gives you 72 factors of 7!

Great question! $7^8 \times 7^9 = 7^{8+9} = 7^{17}$ (you add exponents when multiplying). But $(7^8)^9 = 7^{8 \times 9} = 7^{72}$ (you multiply exponents for a power of a power).

Look for parentheses! If you see $(a^m)^n$ with parentheses, multiply the exponents. If you see $a^m \times a^n$ without parentheses around a power, add the exponents.

The same rule applies! $(x^a)^b = x^{a \times b} = x^{ab}$. You still multiply the exponents, whether they're numbers or variables.

Yes! $(7^8)^9$ means $7^8 \times 7^8 \times 7^8 \times ...$ (9 times). Using the multiplication rule, you'd get $7^{8+8+8+...}$ with 9 eights, which equals $7^{72}$. The power rule is just a shortcut!