Question

write the expression \\(\frac{(2^3)^5}{2^6}\\) as a single power of 2. first, simplify the numerator. \\(\frac{(2^3)^5}{2^6} = \frac{?}{2^6}\\)

Explanation:

Step1: Simplify the numerator

Use exponent power rule: $(a^m)^n = a^{m \times n}$.
For $(2^3)^5$, calculate $3 \times 5 = 15$, so $(2^3)^5 = 2^{15}$.
The expression becomes $\frac{2^{15}}{2^6}$.

Step2: Subtract exponents

Use quotient rule: $\frac{a^m}{a^n} = a^{m-n}$.
Calculate $15 - 6 = 9$, so $\frac{2^{15}}{2^6} = 2^9$.

Answer:

First blank (simplified numerator): $2^{15}$
Final simplified expression: $2^9$