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Find the two square roots of each number. If the number is not a EXERCISES perfect square, approximate the values to one decimal place. (Lesson 1.1) 1. 16 \( \qquad \) 2. \( \frac{4}{25} \) \( \qquad \) 3. 225 \( \qquad \) 4. \( \frac{1}{49} \) \( \qquad \) 5. \( \sqrt{10} \) \( \qquad \) 6. \( \sqrt{18} \) \( \qquad \) Write each decimal as a fraction in simplest form. (Lesson 1.1) 7. \( 0 . \overline{5} \) \( \qquad \) 8. \( 0 . \overline{63} \) \( \qquad \) 9. \( 0 . \overline{214} \) \( \qquad \) Solve each equation for \( x \). (Lesson 1.1) 10. \( x^{2}=361 \) 11. \( x^{3}=1728 \) 12. \( x^{2}=\frac{49}{121} \) \( \qquad \) \( \qquad \) \( \qquad \) Write all names that apply to each number. (Lesson 1.2) 13. \( \frac{2}{3} \) 14. \( -\sqrt{100} \) \( \qquad \) \( \qquad \) 15. \( \frac{15}{5} \) 16. \( \sqrt{21} \) \( \qquad \) \( \qquad \) Compare. Write \( <_{1}> \), or \( = \). (Lesson 1.3) 17. \( \sqrt{7}+5 \) \( 7+\sqrt{5} \) 18. \( 6+\sqrt{8} \) \( \sqrt{6}+8 \) 19. \( \sqrt{4}-2 \) \( 4-\sqrt{2} \) Order the numbers from least to greatest. (Lesson 1.3) 20. \( \sqrt{81}, \frac{72}{7}, 8.9 \) \( \qquad \) 21. \( \sqrt{7}, 2.55, \frac{7}{3} \) \( \qquad \)

Ask by Lewis Watkins. in the United States
Mar 12,2025

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1. \( \sqrt{16} = 4 \) 2. \( \sqrt{\frac{4}{25}} = \frac{2}{5} \) 3. \( \sqrt{225} = 15 \) 4. \( \sqrt{\frac{1}{49}} = \frac{1}{7} \) 5. \( \sqrt{10} \approx 3.162 \) 6. \( \sqrt{18} \approx 4.243 \) 7. \( 0.\overline{5} = \frac{5}{9} \) 8. \( 0.\overline{63} = \frac{63}{99} = \frac{21}{33} = \frac{7}{11} \) 9. \( 0.\overline{214} = \frac{214}{999} \) 10. \( x^2 = 361 \) - \( x = \sqrt{361} = 19 \) or \( x = -19 \) 11. \( x^3 = 1728 \) - \( x = \sqrt[3]{1728} = 12 \) 12. \( x^2 = \frac{49}{121} \) - \( x = \sqrt{\frac{49}{121}} = \frac{7}{11} \) or \( x = -\frac{7}{11} \) 13. \( \frac{2}{3} \) is a **rational number**, **proper fraction**, and **terminating decimal**. 14. \( -\sqrt{100} = -10 \) is a **rational number**, **integer**, and **whole number**. 15. \( \frac{15}{5} = 3 \) is a **rational number**, **integer**, and **whole number**. 16. \( \sqrt{21} \) is an **irrational number**, **real number**, and **positive number**. 17. \( \sqrt{7} + 5 \) vs. \( 7 + \sqrt{5} \) - \( \sqrt{7} \approx 2.6458 \) - \( \sqrt{5} \approx 2.2361 \) - \( \sqrt{7} + 5 \approx 7.6458 \) - \( 7 + \sqrt{5} \approx 9.2361 \) - Therefore, \( \sqrt{7} + 5 < 7 + \sqrt{5} \) 18. \( 6 + \sqrt{8} \) vs. \( \sqrt{6} + 8 \) - \( \sqrt{8} \approx 2.8284 \) - \( \sqrt{6} \approx 2.4495 \) - \( 6 + \sqrt{8} \approx 8.8284 \) - \( \sqrt{6} + 8 \approx 10.4495 \) - Therefore, \( 6 + \sqrt{8} < \sqrt{6} + 8 \) 19. \( \sqrt{4} - 2 \) vs. \( 4 - \sqrt{2} \) - \( \sqrt{4} = 2 \) - \( \sqrt{2} \approx 1.4142 \) - \( \sqrt{4} - 2 = 0 \) - \( 4 - \sqrt{2} \approx 2.5858 \) - Therefore, \( \sqrt{4} - 2 < 4 - \sqrt{2} \) 20. \( \sqrt{81}, \frac{72}{7}, 8.9 \) - \( \sqrt{81} = 9 \) - \( \frac{72}{7} \approx 10.2857 \) - \( 8.9 \) - Therefore, the order from least to greatest is: \( 8.9, \sqrt{81}, \frac{72}{7} \) 21. \( \sqrt{7}, 2.55, \frac{7}{3} \) - \( \sqrt{7} \approx 2.6458 \) - \( \frac{7}{3} \approx 2.3333 \) - \( 2.55 \) - Therefore, the order from least to greatest is: \( \frac{7}{3}, \sqrt{7}, 2.55 \)

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Mind Expander

To start off, let's explore the fascinating historical background of square roots. The concept dates back over 4,000 years to the Babylonians, who used a method akin to a primitive form of our long division to compute square roots. They even had a table for them! Fast forward to the ancient Greeks, and you'll find Euclid’s work on geometry where square roots were already being studied extensively, all the way to the evolution of algebra in the Middle Ages, making this journey through mathematics quite an interesting pursuit. Now, let's have some fun with real-world applications of square roots! From architecture to computer graphics, roots help us find dimensions and unit lengths. For example, in architecture, if you have a square garden of area 100 square feet, taking the square root gives you the length of each side—10 feet! In physics, the square root shows up in calculating the speed of waves, as it influences how we model their behavior. Isn’t it amazing how a simple concept transcends into various aspects of our lives?

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