Solve (3^5)^4: Evaluating Nested Exponent Expression

The rule $(a^n)^m = a^{n \times m}$ comes from repeated multiplication. $(3^5)^4$ means multiply $3^5$ by itself 4 times, which gives us 20 total factors of 3.

Add exponents when multiplying same bases: $3^5 \times 3^4 = 3^{5+4} = 3^9$
Multiply exponents when raising a power to a power: $(3^5)^4 = 3^{5 \times 4} = 3^{20}$

$3^5 \times 3^4 = 3^9$ (adding exponents for multiplication)
$(3^5)^4 = 3^{20}$ (multiplying exponents for nested powers)
The parentheses and outer exponent make all the difference!

Think about what the expression means: $(3^5)^4$ means "take $3^5$ and use it as a base 4 times." Since $3^5$ has 5 factors of 3, doing this 4 times gives 5 × 4 = 20 factors total.

Yes! $(3^5)^4 = 3^5 \times 3^5 \times 3^5 \times 3^5$
Now add the exponents: $3^{5+5+5+5} = 3^{20}$
This shows why we multiply: we're adding the same exponent 4 times!