18. \( \frac{8}{9} \div \frac{1}{3} \) 19. \( \sqrt{\frac{3}{4} \cdot \frac{1}{2}+\frac{3}{16}} \) 20. \( \sqrt{\frac{150}{6}}-\sqrt{\frac{3}{4}-\frac{1}{2}} \)

Calculate the value by following steps: - step0: Calculate: \(\frac{8}{9}\div \frac{1}{3}\) - step1: Multiply by the reciprocal: \(\frac{8}{9}\times 3\) - step2: Reduce the numbers: \(\frac{8}{3}\times 1\) - step3: Multiply: \(\frac{8}{3}\) Calculate or simplify the expression \( \sqrt{\frac{3}{4} \cdot \frac{1}{2}+\frac{3}{16}} \). Calculate the value by following steps: - step0: Calculate: \(\sqrt{\frac{3}{4}\times \frac{1}{2}+\frac{3}{16}}\) - step1: Multiply the numbers: \(\sqrt{\frac{3}{8}+\frac{3}{16}}\) - step2: Add the numbers: \(\sqrt{\frac{9}{16}}\) - step3: Use the properties of radicals: \(\frac{\sqrt{9}}{\sqrt{16}}\) - step4: Simplify the expression: \(\frac{3}{4}\) Calculate or simplify the expression \( \sqrt{\frac{150}{6}}-\sqrt{\frac{3}{4}-\frac{1}{2}} \). Calculate the value by following steps: - step0: Calculate: \(\sqrt{\frac{150}{6}}-\sqrt{\frac{3}{4}-\frac{1}{2}}\) - step1: Divide the terms: \(\sqrt{25}-\sqrt{\frac{3}{4}-\frac{1}{2}}\) - step2: Subtract the numbers: \(\sqrt{25}-\sqrt{\frac{1}{4}}\) - step3: Simplify the root: \(5-\sqrt{\frac{1}{4}}\) - step4: Simplify the root: \(5-\frac{1}{2}\) - step5: Reduce fractions to a common denominator: \(\frac{5\times 2}{2}-\frac{1}{2}\) - step6: Transform the expression: \(\frac{5\times 2-1}{2}\) - step7: Multiply the numbers: \(\frac{10-1}{2}\) - step8: Subtract the numbers: \(\frac{9}{2}\) Let's solve each of the problems step by step. ### Problem 18: \( \frac{8}{9} \div \frac{1}{3} \) To divide fractions, we multiply by the reciprocal of the divisor: \[ \frac{8}{9} \div \frac{1}{3} = \frac{8}{9} \times \frac{3}{1} = \frac{8 \times 3}{9 \times 1} = \frac{24}{9} \] Now, we can simplify \( \frac{24}{9} \): \[ \frac{24}{9} = \frac{8}{3} \quad \text{or} \quad 2\frac{2}{3} \quad \text{or} \quad 2.6\overline{6} \] ### Problem 19: \( \sqrt{\frac{3}{4} \cdot \frac{1}{2}+\frac{3}{16}} \) First, calculate \( \frac{3}{4} \cdot \frac{1}{2} \): \[ \frac{3}{4} \cdot \frac{1}{2} = \frac{3 \times 1}{4 \times 2} = \frac{3}{8} \] Now, add \( \frac{3}{8} \) and \( \frac{3}{16} \). To do this, we need a common denominator, which is 16: \[ \frac{3}{8} = \frac{6}{16} \] Now, add: \[ \frac{6}{16} + \frac{3}{16} = \frac{9}{16} \] Now, take the square root: \[ \sqrt{\frac{9}{16}} = \frac{3}{4} \quad \text{or} \quad 0.75 \] ### Problem 20: \( \sqrt{\frac{150}{6}}-\sqrt{\frac{3}{4}-\frac{1}{2}} \) First, simplify \( \frac{150}{6} \): \[ \frac{150}{6} = 25 \] Now, take the square root: \[ \sqrt{25} = 5 \] Next, simplify \( \frac{3}{4} - \frac{1}{2} \): Convert \( \frac{1}{2} \) to a fraction with a denominator of 4: \[ \frac{1}{2} = \frac{2}{4} \] Now, subtract: \[ \frac{3}{4} - \frac{2}{4} = \frac{1}{4} \] Now, take the square root: \[ \sqrt{\frac{1}{4}} = \frac{1}{2} \] Finally, subtract: \[ 5 - \frac{1}{2} = 5 - 0.5 = 4.5 \quad \text{or} \quad 4\frac{1}{2} \] ### Summary of Results: - **Problem 18:** \( \frac{8}{3} \) or \( 2\frac{2}{3} \) or \( 2.6\overline{6} \) - **Problem 19:** \( \frac{3}{4} \) or \( 0.75 \) - **Problem 20:** \( \frac{9}{2} \) or \( 4\frac{1}{2} \) or \( 4.5 \)