\( \begin{array}{ll}\text { 2. } 16 a^{0} \\ \text { 3. } \frac{2^{-2}}{3^{2}} & \text { 8. } \overline{a^{x}} \\ \text { 4. } \frac{5}{2^{-3}} & \text { 9. } \frac{32 p^{2}}{4 p^{8}} \\ \text { 5. }\left(\frac{2}{3}\right)^{-3} & \text { 10. }\left(2 t^{4}\right)^{3} \\ \left(3^{n+3}\right)^{2} \\ 3^{n} 9^{n-3}\end{array} \)

Sure! Let's simplify each of the given expressions step by step. --- ### **Problem 2.** \( 16 a^{0} \) **Solution:** Any non-zero number raised to the power of 0 is 1. \[ 16 a^{0} = 16 \times 1 = 16 \] **Simplified Form:** \( 16 \) --- ### **Problem 3.** \( \frac{2^{-2}}{3^{2}} \) **Solution:** First, simplify the exponents: \[ 2^{-2} = \frac{1}{2^{2}} = \frac{1}{4} \] \[ 3^{2} = 9 \] Now, divide the two: \[ \frac{2^{-2}}{3^{2}} = \frac{\frac{1}{4}}{9} = \frac{1}{4 \times 9} = \frac{1}{36} \] **Simplified Form:** \( \frac{1}{36} \) --- ### **Problem 4.** \( \frac{5}{2^{-3}} \) **Solution:** Simplify the denominator: \[ 2^{-3} = \frac{1}{2^{3}} = \frac{1}{8} \] Now, divide: \[ \frac{5}{2^{-3}} = 5 \times 2^{3} = 5 \times 8 = 40 \] **Simplified Form:** \( 40 \) --- ### **Problem 5.** \( \left(\frac{2}{3}\right)^{-3} \) **Solution:** A negative exponent indicates the reciprocal: \[ \left(\frac{2}{3}\right)^{-3} = \left(\frac{3}{2}\right)^{3} = \frac{3^{3}}{2^{3}} = \frac{27}{8} \] **Simplified Form:** \( \frac{27}{8} \) --- ### **Problem 8.** \( \overline{a^{x}} \) **Solution:** The notation \( \overline{a^{x}} \) typically represents the complex conjugate of \( a^{x} \) if \( a^{x} \) is a complex number. Without additional context, it cannot be simplified further. **Simplified Form:** \( \overline{a^{x}} \) (Remains as is) --- ### **Problem 9.** \( \frac{32 p^{2}}{4 p^{8}} \) **Solution:** Simplify the coefficients and the exponents separately: \[ \frac{32}{4} = 8 \] \[ \frac{p^{2}}{p^{8}} = p^{2-8} = p^{-6} = \frac{1}{p^{6}} \] Combine them: \[ \frac{32 p^{2}}{4 p^{8}} = 8 \times \frac{1}{p^{6}} = \frac{8}{p^{6}} \] **Simplified Form:** \( \frac{8}{p^{6}} \) --- ### **Problem 10.** \( \left(2 t^{4}\right)^{3} \) **Solution:** Apply the power to both the coefficient and the variable: \[ \left(2 t^{4}\right)^{3} = 2^{3} \times \left(t^{4}\right)^{3} = 8 \times t^{12} = 8 t^{12} \] **Simplified Form:** \( 8 t^{12} \) --- ### **Additional Expressions** #### **(a) \( \left(3^{n+3}\right)^{2} \)** **Solution:** Multiply the exponents: \[ \left(3^{n+3}\right)^{2} = 3^{2(n+3)} = 3^{2n + 6} \] **Simplified Form:** \( 3^{2n + 6} \) --- #### **(b) \( 3^{n} \times 9^{n-3} \)** **Solution:** First, express 9 as a power of 3: \[ 9 = 3^{2} \] \[ 9^{n-3} = \left(3^{2}\right)^{n-3} = 3^{2(n-3)} = 3^{2n - 6} \] Now, multiply the terms with the same base: \[ 3^{n} \times 3^{2n - 6} = 3^{n + 2n - 6} = 3^{3n - 6} \] **Simplified Form:** \( 3^{3n - 6} \) --- Feel free to ask if you need further explanations or assistance with additional problems!