Without using a calculator and by converting decimals into fract simplify the following: \( \begin{array}{ll}\text { (a) } \sqrt{0,64} & \text { (b) } \sqrt[3]{0,008} \\ \text { (c) } \sqrt{2 \frac{7}{9}} & \text { (d) } \frac{\sqrt{64+36}}{\sqrt{64}+\sqrt{36}}\end{array} \)

Let's simplify each of the given expressions by converting decimals to fractions and performing the necessary operations without using a calculator. ### (a) \( \sqrt{0.64} \) 1. **Convert the decimal to a fraction:** \[ 0.64 = \frac{64}{100} = \frac{16}{25} \] 2. **Take the square root:** \[ \sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5} \] **Answer:** \( \frac{4}{5} \) --- ### (b) \( \sqrt[3]{0.008} \) 1. **Convert the decimal to a fraction:** \[ 0.008 = \frac{8}{1000} = \frac{1}{125} \] 2. **Take the cube root:** \[ \sqrt[3]{\frac{1}{125}} = \frac{\sqrt[3]{1}}{\sqrt[3]{125}} = \frac{1}{5} \] **Answer:** \( \frac{1}{5} \) --- ### (c) \( \sqrt{2 \frac{7}{9}} \) 1. **Convert the mixed number to an improper fraction:** \[ 2 \frac{7}{9} = \frac{2 \times 9 + 7}{9} = \frac{25}{9} \] 2. **Take the square root:** \[ \sqrt{\frac{25}{9}} = \frac{\sqrt{25}}{\sqrt{9}} = \frac{5}{3} \] **Answer:** \( \frac{5}{3} \) --- ### (d) \( \frac{\sqrt{64 + 36}}{\sqrt{64} + \sqrt{36}} \) 1. **Simplify the numerator and the denominator separately:** \[ \sqrt{64 + 36} = \sqrt{100} = 10 \] \[ \sqrt{64} = 8 \quad \text{and} \quad \sqrt{36} = 6 \] 2. **Substitute these values back into the expression:** \[ \frac{10}{8 + 6} = \frac{10}{14} = \frac{5}{7} \] **Answer:** \( \frac{5}{7} \) --- ### Summary of Answers: - **(a)** \( \frac{4}{5} \) - **(b)** \( \frac{1}{5} \) - **(c)** \( \frac{5}{3} \) - **(d)** \( \frac{5}{7} \)