Assessment activity 3.1 Simplify the following expressions without using a calculator: \( \begin{array}{ll}\text { 1. } m \times m^{-2} \times m^{4} & \text { 2. } m^{2}+m^{3} \\ x^{-6} \div x^{2} & \text { 4. } x^{3} \div\left(x^{2} \times x\right) \\ \left(x^{3} \div x^{2}\right) \times x & \text { 6. } y \times y^{-2} \\ \left(\frac{x^{4}}{x}\right)^{0} & \text { 8. }\left(x^{2} y^{5}\right)^{0} \\ \left(x^{2} y^{6}\right)^{2} & \text { 10. }\left\{\left(8 x^{3}\right)^{2}\right\}^{\frac{1}{3}} \\ \left\{(-2 x)^{2}\right\}^{3} & \text { 12. }\left\{\left(3 x^{-2}\right)^{2}\right\}^{3} \\ \text { 1. }\left\{\frac{2 x^{-4} y}{2^{-2} x^{3} y^{-2}}\right\}^{\frac{1}{3}} & \text { 14. } 8^{\frac{2}{3}} \\ (16)^{\frac{3}{4}} & \text { 16. }(32)^{\frac{2}{5}} \\ (64)^{-\frac{4}{3}} & \text { 18. }\left(\frac{8}{27}\right)^{-\frac{4}{3}} \\ (-27)^{-\frac{2}{3}} & \text { 20. }\left(\frac{0,09}{0,16}\right)^{-\frac{1}{2}}\end{array} \)

Certainly! Here's the simplification of each expression listed in Assessment Activity 3.1 without using a calculator: --- ### **1. \( m \times m^{-2} \times m^{4} \)** **Simplification:** \[ m^{1} \times m^{-2} \times m^{4} = m^{(1 - 2 + 4)} = m^{3} \] **Answer:** \( m^{3} \) --- ### **2. \( m^{2} + m^{3} \)** **Simplification:** Factor out the common term \( m^{2} \): \[ m^{2} + m^{3} = m^{2}(1 + m) \] **Answer:** \( m^{2}(1 + m) \) --- ### **4. \( x^{-6} \div x^{2} \)** **Simplification:** \[ x^{-6} \div x^{2} = x^{-6 - 2} = x^{-8} = \frac{1}{x^{8}} \] **Answer:** \( \dfrac{1}{x^{8}} \) --- ### **6. \( y \times y^{-2} \)** **Simplification:** \[ y^{1} \times y^{-2} = y^{1 - 2} = y^{-1} = \frac{1}{y} \] **Answer:** \( \dfrac{1}{y} \) --- ### **8. \( (x^{2} y^{5})^{0} \)** **Simplification:** Any non-zero base raised to the power of 0 is 1. \[ (x^{2} y^{5})^{0} = 1 \] **Answer:** \( 1 \) --- ### **10. \( \left\{ \left(8x^{3}\right)^{2} \right\}^{\frac{1}{3}} \)** **Simplification:** \[ (8x^{3})^{2} = 64x^{6} \] \[ \left(64x^{6}\right)^{\frac{1}{3}} = 64^{\frac{1}{3}} \times x^{6 \times \frac{1}{3}} = 4x^{2} \] **Answer:** \( 4x^{2} \) --- ### **12. \( \left\{\left(3x^{-2}\right)^{2}\right\}^{3} \)** **Simplification:** \[ (3x^{-2})^{2} = 9x^{-4} \] \[ (9x^{-4})^{3} = 729x^{-12} = \frac{729}{x^{12}} \] **Answer:** \( \dfrac{729}{x^{12}} \) --- ### **14. \( 8^{\frac{2}{3}} \)** **Simplification:** \[ 8^{\frac{2}{3}} = \left(2^{3}\right)^{\frac{2}{3}} = 2^{2} = 4 \] **Answer:** \( 4 \) --- ### **16. \( (32)^{\frac{2}{5}} \)** **Simplification:** \[ 32 = 2^{5} \] \[ (2^{5})^{\frac{2}{5}} = 2^{2} = 4 \] **Answer:** \( 4 \) --- ### **18. \( \left(\dfrac{8}{27}\right)^{-\frac{4}{3}} \)** **Simplification:** \[ \left(\dfrac{8}{27}\right)^{-\frac{4}{3}} = \left(\dfrac{27}{8}\right)^{\frac{4}{3}} = \left(\dfrac{3^{3}}{2^{3}}\right)^{\frac{4}{3}} = \left(\dfrac{3}{2}\right)^{4} = \dfrac{81}{16} \] **Answer:** \( \dfrac{81}{16} \) --- ### **20. \( \left(\dfrac{0.09}{0.16}\right)^{-\frac{1}{2}} \)** **Simplification:** \[ \left(\dfrac{0.09}{0.16}\right)^{-\frac{1}{2}} = \left(\dfrac{0.16}{0.09}\right)^{\frac{1}{2}} = \sqrt{\dfrac{16}{9}} = \dfrac{4}{3} \] **Answer:** \( \dfrac{4}{3} \) --- These simplifications utilize the properties of exponents and basic algebraic manipulations to arrive at the simplified forms without the need for a calculator.