3. Use the exponent rules to simplify these, and leave your answers in exponential form. \( \begin{array}{ll}\text { a) }\left(8^{6}\right)^{2} & \text { b) }\left(9^{12}\right)^{4} \\ \text { c) }\left(a^{5}\right)^{3} & \text { d) }\left(p^{2}\right)^{2} \\ \text { e) } 2\left(a^{10}\right)^{3} & \text { f) } 3^{2}\left(y^{4}\right)^{3} \\ \text { 4. Determine these values. } & \text { b) } 3^{4} \div 3^{3}+\sqrt[3]{8} \\ \begin{array}{ll}\text { a) }(-4)^{3}+\left(3^{2}\right)^{3} & \text { d) } 3^{10} \div 3^{5} \div 3^{2} \\ \text { c) } 4^{2}+4^{1}+4^{0} & \text { f) } 3 a^{2} \times 2 a^{3} \\ \text { e) } 10^{2} \times 10^{1} \times 10^{0}-10^{3} & \text { b) }\left(3 x^{2}\right)^{3} \\ \text { 5. Simplify. } & \text { d) }\left(\frac{y^{11}}{y^{3}}\right)^{2}\end{array}\end{array} \)

Simplify the expression by following steps: - step0: Solution: \(\left(a^{5}\right)^{3}\) - step1: Multiply the exponents: \(a^{5\times 3}\) - step2: Multiply the numbers: \(a^{15}\) Calculate or simplify the expression \( 3*a^2*2*a^3 \). Simplify the expression by following steps: - step0: Solution: \(3a^{2}\times 2a^{3}\) - step1: Multiply the terms: \(6a^{2}\times a^{3}\) - step2: Multiply the terms: \(6a^{2+3}\) - step3: Add the numbers: \(6a^{5}\) Calculate or simplify the expression \( (8^6)^2 \). Calculate the value by following steps: - step0: Calculate: \(\left(8^{6}\right)^{2}\) - step1: Multiply the exponents: \(8^{6\times 2}\) - step2: Multiply the numbers: \(8^{12}\) Calculate or simplify the expression \( (9^12)^4 \). Calculate the value by following steps: - step0: Calculate: \(\left(9^{12}\right)^{4}\) - step1: Multiply the exponents: \(9^{12\times 4}\) - step2: Multiply the numbers: \(9^{48}\) Calculate or simplify the expression \( (p^2)^2 \). Simplify the expression by following steps: - step0: Solution: \(\left(p^{2}\right)^{2}\) - step1: Multiply the exponents: \(p^{2\times 2}\) - step2: Multiply the numbers: \(p^{4}\) Calculate or simplify the expression \( 3^4/3^3+\sqrt(8) \). Calculate the value by following steps: - step0: Calculate: \(\frac{3^{4}}{3^{3}}+\sqrt{8}\) - step1: Multiply by \(a^{-n}:\) \(3^{4}\times 3^{-3}+\sqrt{8}\) - step2: Simplify the root: \(3^{4}\times 3^{-3}+2\sqrt{2}\) - step3: Multiply the numbers: \(3+2\sqrt{2}\) Calculate or simplify the expression \( 2*(a^10)^3 \). Simplify the expression by following steps: - step0: Solution: \(2\left(a^{10}\right)^{3}\) - step1: Multiply the exponents: \(2a^{10\times 3}\) - step2: Multiply the numbers: \(2a^{30}\) Calculate or simplify the expression \( 3^2*(y^4)^3 \). Simplify the expression by following steps: - step0: Solution: \(3^{2}\left(y^{4}\right)^{3}\) - step1: Multiply the exponents: \(3^{2}y^{4\times 3}\) - step2: Multiply the numbers: \(3^{2}y^{12}\) - step3: Evaluate the power: \(9y^{12}\) Calculate or simplify the expression \( (-4)^3+(3^2)^3 \). Calculate the value by following steps: - step0: Calculate: \(\left(-4\right)^{3}+\left(3^{2}\right)^{3}\) - step1: Multiply the exponents: \(\left(-4\right)^{3}+3^{2\times 3}\) - step2: Multiply the numbers: \(\left(-4\right)^{3}+3^{6}\) - step3: Simplify: \(-4^{3}+3^{6}\) - step4: Evaluate the power: \(-64+3^{6}\) - step5: Evaluate the power: \(-64+729\) - step6: Add the numbers: \(665\) Calculate or simplify the expression \( 4^2+4^1+4^0 \). Calculate the value by following steps: - step0: Calculate: \(4^{2}+4^{1}+4^{0}\) - step1: Evaluate the power: \(4^{2}+4^{1}+1\) - step2: Evaluate the power: \(4^{2}+4+1\) - step3: Evaluate the power: \(16+4+1\) - step4: Add the numbers: \(21\) Calculate or simplify the expression \( 3^10/3^5/3^2 \). Calculate the value by following steps: - step0: Calculate: \(\frac{\frac{3^{10}}{3^{5}}}{3^{2}}\) - step1: Multiply by \(a^{-n}:\) \(\frac{3^{10}\times 3^{-5}}{3^{2}}\) - step2: Multiply by \(a^{-n}:\) \(3^{10}\times 3^{-5}\times 3^{-2}\) - step3: Multiply the terms: \(3^{10-5-2}\) - step4: Subtract the numbers: \(3^{3}\) - step5: Evaluate the power: \(27\) Calculate or simplify the expression \( (3*x^2)^3 \). Simplify the expression by following steps: - step0: Solution: \(\left(3x^{2}\right)^{3}\) - step1: Use the properties of exponents: \(3^{3}\left(x^{2}\right)^{3}\) - step2: Evaluate the power: \(27x^{6}\) Calculate or simplify the expression \( 10^2*10^1*10^0-10^3 \). Calculate the value by following steps: - step0: Calculate: \(10^{2}\times 10^{1}\times 10^{0}-10^{3}\) - step1: Evaluate the power: \(10^{2}\times 10^{1}\times 1-10^{3}\) - step2: Evaluate the power: \(10^{2}\times 10\times 1-10^{3}\) - step3: Multiply the terms: \(10^{3}-10^{3}\) - step4: Subtract the terms: \(0\) Calculate or simplify the expression \( (y^11/y^3)^2 \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{y^{11}}{y^{3}}\right)^{2}\) - step1: Divide the terms: \(\left(y^{8}\right)^{2}\) - step2: Transform the expression: \(y^{8\times 2}\) - step3: Multiply the numbers: \(y^{16}\) Let's simplify each expression step by step using the exponent rules. ### 3. Simplification of Exponential Expressions **a)** \((8^{6})^{2}\) Using the power of a power rule: \((a^m)^n = a^{m \cdot n}\) \[ (8^{6})^{2} = 8^{6 \cdot 2} = 8^{12} \] **b)** \((9^{12})^{4}\) Using the same rule: \[ (9^{12})^{4} = 9^{12 \cdot 4} = 9^{48} \] **c)** \((a^{5})^{3}\) \[ (a^{5})^{3} = a^{5 \cdot 3} = a^{15} \] **d)** \((p^{2})^{2}\) \[ (p^{2})^{2} = p^{2 \cdot 2} = p^{4} \] **e)** \(2(a^{10})^{3}\) \[ 2(a^{10})^{3} = 2a^{10 \cdot 3} = 2a^{30} \] **f)** \(3^{2}(y^{4})^{3}\) \[ 3^{2}(y^{4})^{3} = 3^{2}y^{4 \cdot 3} = 9y^{12} \] ### 4. Determining Values **a)** \((-4)^{3} + (3^{2})^{3}\) Calculating each part: \[ (-4)^{3} = -64 \quad \text{and} \quad (3^{2})^{3} = 9^{3} = 729 \] Thus, \[ -64 + 729 = 665 \] **b)** \(3^{4} \div 3^{3} + \sqrt[3]{8}\) Using the quotient rule: \[ 3^{4} \div 3^{3} = 3^{4-3} = 3^{1} = 3 \] And \(\sqrt[3]{8} = 2\), so: \[ 3 + 2 = 5 \] **c)** \(4^{2} + 4^{1} + 4^{0}\) Calculating each term: \[ 4^{2} = 16, \quad 4^{1} = 4, \quad 4^{0} = 1 \] Thus, \[ 16 + 4 + 1 = 21 \] **d)** \(3^{10} \div 3^{5} \div 3^{2}\) Using the quotient rule: \[ 3^{10} \div 3^{5} = 3^{10-5} = 3^{5} \] Then, \[ 3^{5} \div 3^{2} = 3^{5-2} = 3^{3} = 27 \] **e)** \(10^{2} \times 10^{1} \times 10^{0} - 10^{3}\) Using the product rule: \[ 10^{2} \times 10^{1} \times 10^{0} = 10^{2+1+0} = 10^{3} \] Thus, \[ 10^{3} - 10^{3} = 0 \] **f)** \(3a^{2} \times 2a^{3}\) Using the product rule: \[ 3a^{2} \times 2a^{3} = (3 \times 2)(a^{2} \times a^{3}) = 6a^{2+3} = 6a^{5} \] ### 5. Further Simplification **a)** \((3x^{2})^{3}\) Using the power of a product rule: \[ (3x^{2})^{3} = 3^{3}(x^{2})^{3} = 27x^{6} \] **b)** \(\left(\frac{y^{11}}{y^{3}}\right)^{2}\) Using the quotient rule: \[ \left(\frac{y^{11}}{y^{3}}\right)^{2} = \left(y^{11-3}\right)^{2} = (y^{8})^{2} = y^{16} \] ### Summary of Results - **3a)** \(8^{12}\) - **3b)** \(9^{48}\) - **3c)** \(a^{15}\) - **3d)** \(p^{4}\) - **3e)** \(2a^{30}\) - **3f)** \(9y^{12}\) - **4a)** \(665\) - **4b)** \(5\) - **4c)** \(21\) - **4d)** \(27\) - **4e)** \(0\) - **4f)** \(6a^{5}\) - **5a)** \(27x^{6}\) - **5b)** \(y^{16}\)