Simplify the expressions \( 1 \quad \frac{1}{4} a^{5} b^{12} \times \frac{16}{5} a^{3} b^{2} \) (3) \( \frac{6 a^{4}+9 a^{4}}{5 a^{2}} \) \( 5 \quad p^{x-1} p^{x+1} \) 7 \( 16^{4 x} \div 4^{2 x} \) 9 \( 2^{2010} \times 5^{2011} \)

Let’s simplify those expressions, shall we? 1. To simplify \( \frac{1}{4} a^{5} b^{12} \times \frac{16}{5} a^{3} b^{2} \), multiply the coefficients: \( \frac{1 \times 16}{4 \times 5} = \frac{16}{20} = \frac{4}{5} \). For the variables, add their exponents: \( a^{5+3} = a^{8} \) and \( b^{12+2} = b^{14} \). So, the expression simplifies to \(\frac{4}{5} a^{8} b^{14}\). 2. For \( \frac{6 a^{4}+9 a^{4}}{5 a^{2}} \), start by combining like terms in the numerator: \( 6 a^{4} + 9 a^{4} = 15 a^{4} \). Thus, the expression becomes \( \frac{15 a^{4}}{5 a^{2}} \). Simplifying gives us \( 3 a^{2} \) because \( \frac{15}{5} = 3 \) and \( a^{4-a^{2}} = a^{2} \). 3. To simplify \( 5 \quad p^{x-1} p^{x+1} \), apply the product of powers property, which states that \( p^{a} \times p^{b} = p^{a+b} \). Here, \( p^{(x-1)+(x+1)} = p^{x-1+x+1} = p^{2x} \). 4. For \( 16^{4 x} \div 4^{2 x} \), rewrite \( 16 \) as \( 4^{2} \), resulting in \( (4^{2})^{4x} = 4^{8x} \). Thus, our expression changes to \( \frac{4^{8x}}{4^{2x}} = 4^{8x-2x} = 4^{6x} \). 5. Finally, to simplify \( 2^{2010} \times 5^{2011} \), we’ll keep it as is since there are no common bases to combine. This expression is already in its simplest form: \( 2^{2010} \times 5^{2011} \). So, there you have it! Those expressions have been transformed into their simplest forms.