EXERCISE 1 implify the expressions without using a calculator: \[ \begin{array}{l} \frac{1}{4} a^{5} b^{12} \times \frac{16}{5} a^{3} b^{2} \\ \frac{6 a^{4}+9 a^{4}}{5 a^{2}} \\ p^{x-1} p^{x+1} \\ 16^{4 x} \div 4^{2 x} \\ 2^{2010} \times 5^{2011} \end{array} \] \( 2 \frac{2\left(2 a^{4}\right)^{3}}{16 a^{12}} \) \( 4 \quad \frac{3 a^{3} \times 5 a^{5}}{\left(5 a^{3}\right)^{2}} \) \( 6-2\left(-4 a^{2}+b\right. \) \( 8 \frac{12^{x+1} .27^{x-2}}{18^{2 x-1}} \) \( 10 \quad \frac{3^{x}+3^{x+1}}{8} \)

To tackle these expressions, let's break them down step by step. First up, for \(\frac{1}{4} a^{5} b^{12} \times \frac{16}{5} a^{3} b^{2}\): you can multiply the coefficients, so \(\frac{1}{4} \times \frac{16}{5} = \frac{16}{20} = \frac{4}{5}\), and for the variables, combine the exponents: \(a^{5+3} = a^{8}\) and \(b^{12+2} = b^{14}\). So the simplified expression is \(\frac{4}{5} a^{8} b^{14}\). Next, for \(\frac{6 a^{4}+9 a^{4}}{5 a^{2}}\): combine like terms in the numerator to get \(\frac{15 a^{4}}{5 a^{2}} = 3 a^{2}\) (since \(a^{4} \div a^{2} = a^{2}\)). Now for \(p^{x-1} p^{x+1}\): you can add the exponents, resulting in \(p^{(x-1)+(x+1)} = p^{2x}\). Moving on to \(16^{4 x} \div 4^{2 x}\): convert \(16\) to \(4^2\) which gives \((4^2)^{4x} = 4^{8x}\). Thus, \(\frac{4^{8x}}{4^{2x}} = 4^{8x-2x} = 4^{6x}\). Then, for \(2^{2010} \times 5^{2011}\), there's no simplification applied; leave it as is. Finally, simplify \(2 \frac{2\left(2 a^{4}\right)^{3}}{16 a^{12}}\): First, compute \((2 a^{4})^{3} = 8 a^{12}\), leading to \(\frac{2 \cdot 8 a^{12}}{16 a^{12}} = \frac{16 a^{12}}{16 a^{12}} = 1\). For \(4 \quad \frac{3 a^{3} \times 5 a^{5}}{\left(5 a^{3}\right)^{2}}\): expand to get \(\frac{15 a^{8}}{25 a^{6}} = \frac{3}{5} a^{2}\). For \(6 - 2\left(-4 a^{2}+b\right)\): simplify to \(6 + 8 a^{2} - 2b\). For \(8 \frac{12^{x+1} \cdot 27^{x-2}}{18^{2 x-1}}\): express everything in terms of primes to obtain \(8 \frac{(2^2 \cdot 3)^{x+1} \cdot (3^3)^{x-2}}{(2 \cdot 3^2)^{2x-1}}\) and simplify. Lastly, for \(10 \quad \frac{3^{x}+3^{x+1}}{8}\): factor as \(\frac{3^{x}(1 + 3)}{8} = \frac{4 \cdot 3^x}{8} = \frac{1}{2} \cdot 3^x\). So, there you have your simplified expressions! Happy simplifying!