Simplify the following. leaving your answer in i \( \begin{array}{ll}\text { (a) } 5^{3} \times 5^{4} & \text { (b) } 2^{3+1} \times 2^{-3} \\ \text { (d) } 2^{3}+2^{4} & \text { (c) } 3^{5}+3^{8} \times 3^{4} \\ \text { g) }\left(8^{2}\right)^{3} & \text { (b) }\left(\frac{7^{3}}{5^{3}}\right)^{2}\end{array} \)

**(a)** We use the rule \( a^m \times a^n = a^{m+n} \). \[ 5^3 \times 5^4 = 5^{3+4} = 5^7. \] **(b)** Here, first simplify the exponent in the first factor: \[ 2^{3+1} \times 2^{-3} = 2^4 \times 2^{-3}. \] Then apply the exponent multiplication rule: \[ 2^4 \times 2^{-3} = 2^{4-3} = 2^1 = 2. \] **(d)** The expression is \[ 2^3 + 2^4. \] Notice that \(2^3\) is a common factor: \[ 2^3 + 2^4 = 2^3\left(1 + 2\right) = 2^3 \times 3. \] **(c)** The expression is \[ 3^5 + 3^8 \times 3^4. \] Simplify the product using the rule \( a^m \times a^n = a^{m+n} \): \[ 3^8 \times 3^4 = 3^{8+4} = 3^{12}. \] Thus, the expression becomes: \[ 3^5 + 3^{12}. \] We can factor out the smallest power, \(3^5\): \[ 3^5 + 3^{12} = 3^5\left(1 + 3^{7}\right). \] **(g)** The expression is: \[ \left(8^2\right)^3. \] Apply the power rule \((a^m)^n = a^{mn}\): \[ \left(8^2\right)^3 = 8^{2 \times 3} = 8^6. \] Optionally, expressing in terms of \(2\) since \(8 = 2^3\): \[ 8^6 = \left(2^3\right)^6 = 2^{18}. \] **(b) (Second Instance)** The expression is: \[ \left(\frac{7^3}{5^3}\right)^2. \] Apply the power rule to both numerator and denominator: \[ \left(\frac{7^3}{5^3}\right)^2 = \frac{7^{3 \times 2}}{5^{3 \times 2}} = \frac{7^6}{5^6}. \]