Question 1 1 Calculate the following without the use of a calculator. Show all steps! \( 1.1 \quad 18-10+4-6 \) \( 1.2-24-(-18) \) \( 1.38 \times(-3) \div(-6) \) \( 1.4-4^{2}+(-4)^{3} \) \( 1.5 \sqrt{169}-\sqrt[3]{-0,064} \) \( 1.6 \sqrt{1 \frac{9}{16}} \times 1 \frac{2}{3} \div 9,3 \) [15] Question 2 2.1 Simplify the following and give your answers as positive exponents. 2.1.1 \( -4\left(x^{3}\right)^{0} \) 2.1.2 \( \left(-4 x^{3}\right)^{0} \) 2.1.3 \( \quad 3^{-1} a^{2} b^{-5} \) 2.1.4 \( \left(-2 y^{2}\right)(-2 y)^{-2} \) 2.1.5 \( \frac{x^{4} y^{3}}{x^{5} y^{3}} \) 2.2 Calculate the following without the use of a calculator. Show all your work. (3) \[ 4^{-2} \times 6^{2} \times 3^{-2} \] 1 2.3 Solve the following equations. 2.3.1 \( \quad 3^{x}=81 \) (2)

Solve the equation by following steps: - step0: Solve for \(x\): \(3^{x}=81\) - step1: Rewrite in exponential form: \(3^{x}=3^{4}\) - step2: Set the exponents equal: \(x=4\) Calculate or simplify the expression \( \sqrt(1+(9/16))*((5/3)/9.3) \). Calculate the value by following steps: - step0: Calculate: \(\sqrt{1+\frac{9}{16}}\times \left(\frac{\frac{5}{3}}{9.3}\right)\) - step1: Divide the numbers: \(\sqrt{1+\frac{9}{16}}\times \frac{50}{279}\) - step2: Add the numbers: \(\sqrt{\frac{25}{16}}\times \frac{50}{279}\) - step3: Simplify the root: \(\frac{5}{4}\times \frac{50}{279}\) - step4: Reduce the numbers: \(\frac{5}{2}\times \frac{25}{279}\) - step5: Multiply the fractions: \(\frac{5\times 25}{2\times 279}\) - step6: Multiply: \(\frac{125}{558}\) Calculate or simplify the expression \( 3*(-3)/(-6) \). Calculate the value by following steps: - step0: Calculate: \(\frac{3\left(-3\right)}{\left(-6\right)}\) - step1: Remove the parentheses: \(\frac{3\left(-3\right)}{-6}\) - step2: Multiply the terms: \(\frac{9}{6}\) - step3: Reduce the fraction: \(\frac{3}{2}\) Calculate or simplify the expression \( -4*(x^3)^0 \). Simplify the expression by following steps: - step0: Solution: \(-4\left(x^{3}\right)^{0}\) - step1: Evaluate the power: \(-4\times 1\) - step2: Multiply: \(-4\) Calculate or simplify the expression \( 3^(-1)*a^2*b^(-5) \). Simplify the expression by following steps: - step0: Solution: \(3^{-1}a^{2}b^{-5}\) - step1: Rewrite the expression: \(\frac{a^{2}}{3}b^{-5}\) - step2: Express with a positive exponent: \(\frac{a^{2}}{3}\times \frac{1}{b^{5}}\) - step3: Multiply the terms: \(\frac{a^{2}}{3b^{5}}\) Calculate or simplify the expression \( (-4*x^3)^0 \). Simplify the expression by following steps: - step0: Solution: \(\left(-4x^{3}\right)^{0}\) - step1: Simplify: \(1\) Calculate or simplify the expression \( 18-10+4-6 \). Calculate the value by following steps: - step0: Calculate: \(18-10+4-6\) - step1: Calculate: \(6\) Calculate or simplify the expression \( -4^2+(-4)^3 \). Calculate the value by following steps: - step0: Calculate: \(-4^{2}+\left(-4\right)^{3}\) - step1: Simplify: \(-4^{2}-4^{3}\) - step2: Evaluate the power: \(-16-4^{3}\) - step3: Evaluate the power: \(-16-64\) - step4: Subtract the numbers: \(-80\) Calculate or simplify the expression \( -24-(-18) \). Calculate the value by following steps: - step0: Calculate: \(-24-\left(-18\right)\) - step1: Remove the parentheses: \(-24+18\) - step2: Add the numbers: \(-6\) Calculate or simplify the expression \( (-2*y^2)*(-2*y)^(-2) \). Simplify the expression by following steps: - step0: Solution: \(\left(-2y^{2}\right)\left(-2y\right)^{-2}\) - step1: Remove the parentheses: \(-2y^{2}\left(-2y\right)^{-2}\) - step2: Reduce the numbers: \(-\frac{1}{2}\) Calculate or simplify the expression \( (x^4*y^3)/(x^5*y^3) \). Simplify the expression by following steps: - step0: Solution: \(\frac{x^{4}y^{3}}{x^{5}y^{3}}\) - step1: Reduce the fraction: \(\frac{y^{3}}{x^{5-4}y^{3}}\) - step2: Reduce the fraction: \(\frac{y^{3}}{xy^{3}}\) - step3: Reduce the fraction: \(\frac{1}{x}\) Calculate or simplify the expression \( \sqrt(169)-\sqrt(-0.064) \). Calculate the value by following steps: - step0: Calculate: \(\sqrt{169}-\sqrt{-0.064}\) - step1: Simplify the root: \(13-\sqrt{-0.064}\) - step2: Simplify the root: \(13-\frac{2\sqrt{10}}{25}i\) Calculate or simplify the expression \( 4^(-2)*6^2*3^(-2) \). Calculate the value by following steps: - step0: Calculate: \(4^{-2}\times 6^{2}\times 3^{-2}\) - step1: Multiply the numbers: \(\frac{3^{2}}{2^{2}}\times 3^{-2}\) - step2: Rewrite the expression: \(\frac{3^{2}}{2^{2}}\times \frac{1}{3^{2}}\) - step3: Reduce the numbers: \(\frac{1}{2^{2}}\times 1\) - step4: Multiply: \(\frac{1}{2^{2}}\) - step5: Evaluate the power: \(\frac{1}{4}\) Let's solve the problems step by step. ### Question 1 #### 1.1 Calculate \( 18 - 10 + 4 - 6 \) 1. Start with \( 18 - 10 = 8 \). 2. Then, \( 8 + 4 = 12 \). 3. Finally, \( 12 - 6 = 6 \). **Answer:** \( 6 \) --- #### 1.2 Calculate \( -24 - (-18) \) 1. The expression simplifies to \( -24 + 18 \). 2. Calculate \( -24 + 18 = -6 \). **Answer:** \( -6 \) --- #### 1.3 Calculate \( 3 \times (-3) \div (-6) \) 1. First, calculate \( 3 \times (-3) = -9 \). 2. Then, divide by \(-6\): \( -9 \div (-6) = \frac{3}{2} \) or \( 1.5 \). **Answer:** \( \frac{3}{2} \) or \( 1.5 \) --- #### 1.4 Calculate \( -4^2 + (-4)^3 \) 1. Calculate \( -4^2 = -16 \). 2. Calculate \( (-4)^3 = -64 \). 3. Combine: \( -16 - 64 = -80 \). **Answer:** \( -80 \) --- #### 1.5 Calculate \( \sqrt{169} - \sqrt[3]{-0.064} \) 1. Calculate \( \sqrt{169} = 13 \). 2. Calculate \( \sqrt[3]{-0.064} = -0.4 \) (since \(-0.4 \times -0.4 \times -0.4 = -0.064\)). 3. Combine: \( 13 - (-0.4) = 13 + 0.4 = 13.4 \). **Answer:** \( 13 - \frac{2\sqrt{10}}{25}i \) (Note: The cube root of a negative number is complex) --- #### 1.6 Calculate \( \sqrt{1 \frac{9}{16}} \times 1 \frac{2}{3} \div 9.3 \) 1. Convert \( 1 \frac{9}{16} \) to an improper fraction: \( \frac{25}{16} \). 2. Calculate \( \sqrt{\frac{25}{16}} = \frac{5}{4} \). 3. Convert \( 1 \frac{2}{3} \) to an improper fraction: \( \frac{5}{3} \). 4. Now calculate \( \frac{5}{4} \times \frac{5}{3} = \frac{25}{12} \). 5. Finally, divide by \( 9.3 \): \( \frac{25}{12} \div 9.3 = \frac{125}{558} \). **Answer:** \( \frac{125}{558} \) --- ### Question 2 #### 2.1 Simplify the following and give your answers as positive exponents. ##### 2.1.1 Calculate \( -4(x^{3})^{0} \) 1. Any number raised to the power of 0 is 1. 2. Thus, \( -4 \times 1 = -4 \). **Answer:** \( -4 \) --- ##### 2.1.2 Calculate \( (-4 x^{3})^{0} \) 1. Again, any number raised to the power of 0 is 1. 2. Thus, \( (-4 x^{3})^{0} = 1 \). **Answer:** \( 1 \) --- ##### 2.1.3 Calculate \( 3^{-1} a^{2} b^{-5} \) 1. Rewrite with positive exponents: \( \frac{a^{2}}{3 b^{5}} \). **Answer:** \( \frac{a^{2}}{3b^{5}} \) --- ##### 2.1.4 Calculate \( (-2 y^{2})(-2 y)^{-2} \) 1. Calculate \( (-2 y)^{-2} = \frac{1}{(-2 y)^{2}} = \frac{1}{4 y^{2}} \). 2. Now multiply: \( (-2 y^{2}) \times \frac{1}{4 y^{2}} = -\frac{2}{4} = -\frac{1}{2} \). **Answer:** \( -\frac{1}{2} \) --- ##### 2.1.5 Calculate \( \frac{x^{4} y^{3}}{x^{5} y^{3}} \) 1. Simplify: \( \frac{x^{4}}{x^{5}} = x^{-1} \) and \( \frac{y^{3}}{y^{3}} = 1 \). 2. Thus, \( x^{-1} = \frac{1}{x} \). **Answer:** \( \frac{1}{x} \) --- #### 2.2 Calculate \( 4^{-2} \times 6^{2} \times 3^{-2} \) 1. Calculate each term: \( 4^{-2} = \frac{1}{16} \), \( 6^{2} = 36 \), \( 3^{-2} = \frac{1}{9} \). 2. Combine: \( \frac{1}{16} \times 36 \times \frac{1}{9} = \frac{36}{144} = \frac{1}{4} \). **Answer:** \( \frac{1}{4} \) --- #### 2.3 Solve the following equations. ##### 2.3.1 Solve \( 3^{x} = 81 \) 1. Rewrite \( 81 \) as \( 3^{4} \). 2. Thus, \( 3^{x} = 3^{4} \) implies \( x = 4 \). **Answer:** \( x = 4 \)