(b) Simplify the following: \( \begin{array}{lll}\text { (1) } 2^{x} \cdot 2^{x} & \text { (2) }\left(2^{5}\right)^{x} & \text { (3) }\left(3^{x}\right)^{3} \\ \text { (4) } 2^{x} \cdot 2^{3} & \text { (5) }\left(5^{3} \cdot 3^{5}\right)^{x} & \text { (6) } 9^{x} \cdot 3^{2 x} \\ \text { (7) } 81^{x} \cdot 27^{2 x} & \text { (8) } \frac{5^{x} \cdot 5^{x}}{25^{x}} & \text { (9) } \frac{\left(2^{x}\right)^{3} \cdot 2^{x}}{16^{x}} \\ \text { (10) } \frac{7^{x} \cdot 7^{2 x}}{\left(7^{2} \cdot 7\right)^{x}} & \text { (11) } \frac{25 \cdot 9^{x}}{3^{x} \cdot 3^{x} \cdot 5} & \text { (12) } \frac{\left(2^{x}\right)^{3} \cdot 27^{x}}{8^{x} \cdot\left(3^{2}\right)^{x} \cdot 3^{x}} \\ \text { (13) } \frac{12^{x} \cdot 3^{-x}}{2 \cdot 4^{x}} & \text { (14) } \frac{32^{x}}{4^{x}} & \end{array} \)

We wish to “simplify” each expression so that the answer is written in a single exponential form (or in simplest form). One acceptable set of answers is given below. Let’s go through each one. ───────────────────────────── (1) Simplify 2ˣ · 2ˣ When multiplying powers with the same base add the exponents:   2ˣ · 2ˣ = 2^(x + x) = 2^(2x). ───────────────────────────── (2) Simplify (2⁵)ˣ Recall the power rule: (aᵇ)ˣ = a^(b·x), so:   (2⁵)ˣ = 2^(5x). ───────────────────────────── (3) Simplify (3ˣ)³ Similarly, (3ˣ)³ = 3^(3x). ───────────────────────────── (4) Simplify 2ˣ · 2³ Since the bases are the same, add exponents:   2ˣ · 2³ = 2^(x + 3). ───────────────────────────── (5) Simplify (5³ · 3⁵)ˣ Use the rule (ab)ˣ = aˣ · bˣ:   (5³ · 3⁵)ˣ = (5³)ˣ · (3⁵)ˣ = 5^(3x) · 3^(5x). ───────────────────────────── (6) Simplify 9ˣ · 3^(2x) Since 9 = 3², write:   9ˣ = (3²)ˣ = 3^(2x). Then   9ˣ · 3^(2x) = 3^(2x) · 3^(2x) = 3^(2x+2x) = 3^(4x). ───────────────────────────── (7) Simplify 81ˣ · 27^(2x) Express 81 and 27 as powers of 3:   81 = 3⁴ and 27 = 3³. So,   81ˣ = (3⁴)ˣ = 3^(4x),   27^(2x) = (3³)^(2x) = 3^(6x). Thus,   81ˣ · 27^(2x) = 3^(4x+6x) = 3^(10x). ───────────────────────────── (8) Simplify (5ˣ · 5ˣ) / 25ˣ Write numerator: 5ˣ · 5ˣ = 5^(2x). Write 25 in terms of 5: 25 = 5² so   25ˣ = (5²)ˣ = 5^(2x). Thus,   (5^(2x))/(5^(2x)) = 1. ───────────────────────────── (9) Simplify [(2ˣ)³ · 2ˣ] / 16ˣ Rewrite numerator:   (2ˣ)³ = 2^(3x), so numerator becomes 2^(3x) · 2ˣ = 2^(3x + x) = 2^(4x). Write 16 in terms of 2: 16 = 2⁴ so 16ˣ = (2⁴)ˣ = 2^(4x). Then the expression is 2^(4x) / 2^(4x) = 1. ───────────────────────────── (10) Simplify (7ˣ · 7^(2x)) / ((7² · 7)ˣ) Combine the numerator:   7ˣ · 7^(2x) = 7^(x+2x) = 7^(3x). For the denominator, note:   7² · 7 = 7^(2+1)=7³, so (7² · 7)ˣ = (7³)ˣ = 7^(3x). Thus, the expression becomes 7^(3x)/7^(3x) = 1. ───────────────────────────── (11) Simplify (25 · 9ˣ) / (3ˣ · 3ˣ · 5) Combine the 3ˣ’s in the denominator: 3ˣ · 3ˣ = 3^(2x). Also, note 9 = 3² so 9ˣ = (3²)ˣ = 3^(2x). Then we have:   (25 · 3^(2x)) / (5 · 3^(2x)). Cancel 3^(2x) (provided 3^(2x) ≠ 0) to obtain:   25/5 = 5. ───────────────────────────── (12) Simplify [(2ˣ)³ · 27ˣ] / (8ˣ · (3²)ˣ · 3ˣ) Write (2ˣ)³ = 2^(3x). Also, note 27 = 3³ so 27ˣ = (3³)ˣ = 3^(3x). In the denominator, 8 = 2³ so 8ˣ = (2³)ˣ = 2^(3x); also, (3²)ˣ = 3^(2x) and then extra factor 3ˣ gives 3^(2x+x)=3^(3x). Thus:   Numerator = 2^(3x) · 3^(3x) and denominator = 2^(3x) · 3^(3x), so the fraction is 1. ───────────────────────────── (13) Simplify (12ˣ · 3^(–x)) / (2 · 4ˣ) A good strategy is to express 12 as 3·4. Notice that:   12ˣ = (3·4)ˣ = 3ˣ · 4ˣ. Thus numerator becomes: 3ˣ · 4ˣ · 3^(–x) = (3ˣ · 3^(–x)) · 4ˣ = 3^(0) · 4ˣ = 4ˣ. Denominator is 2 · 4ˣ. Cancel 4ˣ (assuming 4ˣ ≠ 0) giving 1/2. ───────────────────────────── (14) Simplify 32ˣ / 4ˣ Write 32 = 2⁵ and 4 = 2². Then:   32ˣ = (2⁵)ˣ = 2^(5x) and 4ˣ = (2²)ˣ = 2^(2x). Thus,   32ˣ / 4ˣ = 2^(5x)/2^(2x) = 2^(5x – 2x) = 2^(3x). ───────────────────────────── Final Answers (1) 2^(2x) (2) 2^(5x) (3) 3^(3x) (4) 2^(x+3) (5) 5^(3x) · 3^(5x) (6) 3^(4x) (7) 3^(10x) (8) 1 (9) 1 (10) 1 (11) 5 (12) 1 (13) 1/2 (14) 2^(3x) Any answer written in an equivalent form is correct.