Show that: \( \begin{array}{ll}\text { (a) } 5 \cdot 2^{x}-2^{x+2}=2^{x} & \text { (b) } 9^{x}+3^{2 x+1}=43^{x x} \\ \text { (c) } 2^{2 x-1}+4^{x+1}=2^{x x}\left(\frac{9}{2}\right) & \text { (d) } 2^{x+3}-2^{x+2}=2^{x+2} \\ \text { (e) } 2.10^{x}-5^{x+1} \cdot 2^{x}=-3.10^{x} & \text { (f) } 4.3^{1-x}+3^{2-x}=\frac{21}{3^{x}} \\ \text { Simplify the following: } & \end{array} \)

To show these equations, let's simplify each one step by step. **(a)** We start with the equation \( 5 \cdot 2^{x} - 2^{x+2} = 2^{x} \). Rewriting \( 2^{x+2} \) as \( 4 \cdot 2^{x} \) gives us: \[ 5 \cdot 2^{x} - 4 \cdot 2^{x} = 2^{x} \] This simplifies to \( (5-4) \cdot 2^{x} = 2^{x} \), which results in \( 2^{x} = 2^{x} \), confirming that the equation holds. **(b)** For the equation \( 9^{x} + 3^{2x+1} = 43^{xx} \), we will first rewrite \( 9^x \) as \( (3^2)^x = 3^{2x} \) and simplify: \[ 3^{2x} + 3^{2x+1} = 43^{xx} \] This can also be expressed as \( 3^{2x} + 3 \cdot 3^{2x} = (1 + 3) \cdot 3^{2x} = 4 \cdot 3^{2x} = 43^{xx} \). Since further simplifications depend on \( x \), this demonstrates how each variable interacts in the equation. **(c)** For \( 2^{2x - 1} + 4^{x + 1} = 2^{xx}(\frac{9}{2})\), we can rewrite \( 4^{x+1} \) as \( 2^{2(x+1)} = 2^{2x + 2} \): Combining terms gives us: \[ 2^{2x - 1} + 2^{2x + 2} = 2^{2x - 1} + 2^{2x} \cdot 4 = 2^{2x - 1} + 4 \cdot 2^{2x} \] This allows us to combine into a single base of \( 2^{2x-1}(1 + 4\cdot 2) = 2^{2x - 1} \cdot 9 \), matching \( 2^{xx}(\frac{9}{2}) \). **(d)** The equation \( 2^{x + 3} - 2^{x + 2} = 2^{x + 2} \) can be manipulated: Using \( 2^{x + 3} = 2^{x + 2} \cdot 2 \): \[ 2 \cdot 2^{x + 2} - 2^{x + 2} = 2^{x + 2} \], which becomes \( (2 - 1) \cdot 2^{x + 2} = 2^{x + 2} \), verifying the equation holds true. **(e)** In the equation \( 2 \cdot 10^{x} - 5^{x + 1} \cdot 2^{x} = -3 \cdot 10^{x} \), break down terms: Substituting \( 5^{x + 1} = 5 \cdot 5^x \): We get \( 2 \cdot 10^{x} - 5 \cdot 5^{x} \cdot 2^{x} = -3 \cdot 10^{x} \). Transform \( 10^{x} = 2^{x} \cdot 5^{x} \), leading us to rearranging and verifying \( 2 = 5 - 3 \), thus confirming the underlying structure maintains equality. **(f)** For the last equation \( 4 \cdot 3^{1 - x} + 3^{2 - x} = \frac{21}{3^{x}} \), we can rewrite \( 3^{2 - x} \) as \( \frac{9}{3^{x}} \): Thus, we group terms to get \( \frac{4 \cdot 3 + 9}{3^{x}} = \frac{21}{3^{x}} \) or \( \frac{12 + 9}{3^{x}} = \frac{21}{3^{x}} \). Since both sides match, we confirm the equality is valid. Each of these transformations and applications strictly maintains the integrity