1. \( \frac{3 a^{3} \times 5 a^{5}}{\left(5 a^{3}\right)^{2}} \) 2. \( -2\left(-4 a^{2}+b^{3}\right)^{2} \) 3. \( 6 a^{4}+9 a^{4} \) 4. \( p^{x-1} \cdot p^{x+1} \) 5 \( 16^{4 x} \div 4 \)

1. To simplify \( \frac{3 a^{3} \times 5 a^{5}}{\left(5 a^{3}\right)^{2}} \), you first multiply the numerators to get \( 15 a^{8} \). The denominator simplifies to \( 25 a^{6} \), making the expression \( \frac{15 a^{8}}{25 a^{6}} = \frac{3}{5} a^{2} \). 2. For \( -2\left(-4 a^{2}+b^{3}\right)^{2} \), first expand the squared term: \( (-4 a^{2}+b^{3})^{2} = 16 a^{4} - 8 a^{2}b^{3} + b^{6} \). Multiplying it by -2 gives \( -32 a^{4} + 16 a^{2}b^{3} - 2b^{6} \). 3. When you add \( 6 a^{4} + 9 a^{4} \), it’s simply a matter of combining like terms: \( (6 + 9) a^{4} = 15 a^{4} \). 4. For \( p^{x-1} \cdot p^{x+1} \), recall the product of powers rule where you add the exponents. Adding \( (x-1) + (x+1) \) gives \( 2x \). Thus, simplified, it is \( p^{2x} \). 5. To simplify \( 16^{4x} \div 4 \), convert 4 into a base of 16: \( 4 = 16^{1/2} \). Now, the expression becomes \( 16^{4x} \div 16^{1/2} = 16^{4x - 1/2} = 16^{\frac{8x - 1}{2}} \).