1. Problem: Simplify the expression $4x^5 \times (2x)^2 \div 8x$. 2. Formula and rules: - When multiplying powers with the same base, add exponents: $a^m \times a^n = a^{m+n}$. - When dividing powers with the same base, subtract exponents: $\frac{a^m}{a^n} = a^{m-n}$. - Apply exponent to a product: $(ab)^n = a^n b^n$. 3. Step-by-step simplification: - Rewrite $(2x)^2$ as $2^2 \times x^2 = 4x^2$. - Substitute back: $4x^5 \times 4x^2 \div 8x$. - Multiply numerators: $4 \times 4 = 16$, and $x^5 \times x^2 = x^{5+2} = x^7$. - So numerator is $16x^7$. - Denominator is $8x = 8x^1$. - Write the fraction: $\frac{16x^7}{8x^1}$. - Simplify coefficients: $\frac{16}{8} = \cancel{\frac{16}{8}}{2}$. - Simplify variables: $\frac{x^7}{x^1} = x^{7-1} = x^6$. - Final simplified expression: $2x^6$. Final answer: $2x^6$