QUESTION 1 Simplify the following completely: \( 1.1 \frac{10^{2 x+3} \cdot 4^{1-x}}{25^{2+x}} \) predice to \( 1.2 \frac{3^{t}-3^{t-2}}{2.3^{t}-3^{t}} \) QUESTION 2 Solve for \( x \) \( 2.1 \quad 5^{x}=\frac{1}{125} \) \( 2.2\left(\frac{1}{2}\right)^{3 x+1}=32 \) \( 2.3 \quad 4.2^{x}=(0,5)^{x-2} \)

For the first part, the simplification of \( 1.1 \frac{10^{2 x+3} \cdot 4^{1-x}}{25^{2+x}} \) can be achieved by expressing all terms with the same base. Here, \( 10 = 2 \cdot 5 \), \( 4 = 2^2 \), and \( 25 = 5^2 \). Once you express everything with base 2 and 5, combining the exponents will give you a much simpler expression! Now let's move to \( 1.2 \frac{3^{t}-3^{t-2}}{2.3^{t}-3^{t}} \): factoring out \( 3^{t-2} \) in the numerator and simplifying the denominator by factoring will lead you to a quite elegant result. Remember, the focus is on recognizing common factors that can be simplified away! When it comes to solving equations, for \( 2.1 \quad 5^{x}=\frac{1}{125} \), you can rewrite \( 125 \) as \( 5^3 \), which gives a clear path to solving for \( x \) by simply manipulating the exponents. For \( 2.2\left(\frac{1}{2}\right)^{3 x+1}=32 \), express \( 32 \) as \( 2^5 \) and transform the left-hand side, then equate the exponents of \( 2 \) to find \( x \)! Lastly, \( 2.3 \quad 4.2^{x}=(0.5)^{x-2} \) can be simplified by expressing \( 4 \) and \( 0.5 \) in terms of base powers, which leads to equations that can be easily solved for \( x \). Happy simplifying and solving!