(b) Simplify the following: \( \begin{array}{lll}\text { (1) } 2^{x} \cdot 2^{x} & \text { (2) } & \left(2^{5}\right)^{x} \\ \text { (4) } 2^{x} \cdot 2^{3} & \text { (5) } & \left(5^{3} \cdot 3^{5}\right)^{x} \\ \text { (7) } 81^{x} \cdot 27^{2 x} & \text { (8) } \frac{5^{x} \cdot 5^{x}}{25^{x}} & \text { (3) } \\ \left(3^{x}\right)^{3} \\ 9^{x} \cdot 3^{2 x} \\ \text { (10) } \frac{7^{x} \cdot 7^{2 x}}{\left(7^{2} \cdot 7\right)^{x}} & \text { (11) } \frac{25 \cdot 9^{x}}{3^{x} \cdot 3^{x} \cdot 5} & \text { (9) } \frac{\left(2^{x}\right)^{3} \cdot 2^{x}}{16^{x}} \\ \text { (13) } \frac{12^{x} \cdot 3^{-x}}{2 \cdot 4^{x}} & \text { (14) } \frac{32^{x}}{4^{x}} & \text { (12) } \frac{\left(2^{x}\right)^{3} \cdot 27^{x}}{8^{x} \cdot\left(3^{2}\right)^{x} \cdot 3^{x}}\end{array} \)

We wish to simplify each expression. One acceptable answer for each is shown below. ────────────────────────────── 1.  2ˣ · 2ˣ   • When multiplying with the same base, we add exponents.   • 2ˣ · 2ˣ = 2^(x + x) = 2^(2x). ────────────────────────────── 2.  (2⁵)ˣ   • When raising a power to a power, multiply exponents.   • (2⁵)ˣ = 2^(5x). ────────────────────────────── 3.  (3ˣ)³   • (3ˣ)³ = 3^(x·3) = 3^(3x). ────────────────────────────── 4.  2ˣ · 2³   • Same base: add the exponents.   • 2ˣ · 2³ = 2^(x + 3). ────────────────────────────── 5.  (5³ · 3⁵)ˣ   • When a product is raised to a power, distribute the exponent.   • (5³ · 3⁵)ˣ = 5^(3x) · 3^(5x). ────────────────────────────── 7.  81ˣ · 27^(2x)   • Write each number as a power of 3: 81 = 3⁴ and 27 = 3³.   • Then 81ˣ = (3⁴)ˣ = 3^(4x) and 27^(2x) = (3³)^(2x) = 3^(6x).   • Multiply: 3^(4x) · 3^(6x) = 3^(4x + 6x) = 3^(10x). ────────────────────────────── 8.  (5ˣ · 5ˣ)⁄(25ˣ)   • Multiply numerator: 5ˣ · 5ˣ = 5^(x + x) = 5^(2x).   • But 25 = 5² so 25ˣ = (5²)ˣ = 5^(2x).   • Thus, 5^(2x)/5^(2x) = 1. ────────────────────────────── 9.  [(2ˣ)³ · 2ˣ]⁄(16ˣ)   • (2ˣ)³ = 2^(3x) so the numerator is 2^(3x) · 2ˣ = 2^(4x).   • Since 16 = 2⁴, 16ˣ = (2⁴)ˣ = 2^(4x).   • Therefore 2^(4x)/2^(4x) = 1. ────────────────────────────── 10.  [7ˣ · 7^(2x)]⁄[(7² · 7)ˣ]   • In the numerator, 7ˣ · 7^(2x) = 7^(x + 2x) = 7^(3x).   • In the denominator, 7² · 7 = 7^(2+1) = 7³ so (7² · 7)ˣ = (7³)ˣ = 7^(3x).   • Thus, the expression equals 7^(3x)/7^(3x) = 1. ────────────────────────────── 11.  (25 · 9ˣ)⁄(3ˣ · 3ˣ · 5)   • First, note 3ˣ · 3ˣ = 3^(2x).   • Write 25/5 = 5.   • Also, 9 = 3² so 9ˣ = (3²)ˣ = 3^(2x).   • Then the expression becomes 5 · 3^(2x)/3^(2x) = 5. ────────────────────────────── 12.  [(2ˣ)³ · 27ˣ]⁄[8ˣ · (3²)ˣ · 3ˣ]   • Write (2ˣ)³ = 2^(3x) and 27 = 3³ so 27ˣ = 3^(3x).   • Write 8 = 2³ so 8ˣ = 2^(3x).   • Also, (3²)ˣ = 3^(2x). Then multiplying denominator factors: 3^(2x) · 3ˣ = 3^(3x).   • So the expression is (2^(3x) · 3^(3x))/(2^(3x) · 3^(3x)) = 1. ────────────────────────────── 13.  (12ˣ · 3^(–x))⁄(2 · 4ˣ)   • Write 12 = 2² · 3 so 12ˣ = (2² · 3)ˣ = 2^(2x) · 3ˣ.   • Then the numerator becomes 2^(2x) · 3ˣ · 3^(–x) = 2^(2x) (since 3ˣ · 3^(–x)=3^(0)=1).   • Write 4 = 2² so 4ˣ = 2^(2x).   • Denominator now is 2 · 2^(2x) = 2^(2x+1).   • Thus, the expression is 2^(2x)/2^(2x+1) = 1/2. ────────────────────────────── 14.  32ˣ⁄4ˣ   • Write 32 = 2⁵ and 4 = 2².   • Then 32ˣ = 2^(5x) and 4ˣ = 2^(2x).   • Thus, the expression is 2^(5x)/2^(2x) = 2^(3x). ────────────────────────────── Summarizing the answers:  (1) 2^(2x)  (2) 2^(5x)  (3) 3^(3x)  (4) 2^(x+3)  (5) 5^(3x) · 3^(5x)  (7) 3^(10x)  (8) 1  (9) 1  (10) 1  (11) 5  (12) 1  (13) ½  (14) 2^(3x) Any answer equivalent to these is correct.