1. **Problem:** The side length of a square is $\sqrt{27}$ inches. Select all expressions equal to the area of this square. 2. **Formula:** The area $A$ of a square with side length $s$ is given by: $$A = s^2$$ 3. **Calculate the area:** $$A = (\sqrt{27})^2$$ 4. **Simplify:** $$A = 27$$ 5. **Check each expression:** - a) 5.2 (approximate, but $\sqrt{27} \approx 5.196$, so $5.2$ is close to side length, not area) - b) $27^2 = 729$ (not equal to area) - c) 27 (equal to area) - d) $\sqrt{27}$ (side length, not area) - e) $(\sqrt{27})^2 = 27$ (equal to area) **Answer:** c) and e) are equal to the area. 2. **Problem:** Determine the area of the square with side length $\sqrt{27}$. **Answer:** $27$ square inches. 3. **Problem:** Determine the exact side length (in radical form) of the shaded square. **Answer:** $\sqrt{27}$ inches (given). 4. **Problem:** Find the exact solution to $x^2 = 34$. **Formula:** $$x = \pm \sqrt{34}$$ **Answer:** d) $x = \sqrt{34}$ 5. **Problem:** Square A has area 81 square feet. Select all expressions equal to the side length. **Formula:** $$s = \sqrt{\text{area}} = \sqrt{81} = 9$$ **Check:** - a) 3 (no, $3^2=9$) - b) $81/2 = 40.5$ (no) - c) $\sqrt{81} = 9$ (yes) - d) $\sqrt{9} = 3$ (no) - e) 9 (yes) **Answer:** c) and e) 6. **Problem:** Select all numbers greater than 7 and less than 8. Calculate approximate values: - $\sqrt{19} \approx 4.36$ (no) - $\sqrt{36} = 6$ (no) - $\sqrt{50} \approx 7.07$ (yes) - $\sqrt{62} \approx 7.87$ (yes) - $\sqrt{64} = 8$ (no, equals 8) **Answer:** c) and d) 7. **Problem:** Rectangular field with width 80 yards and length 115 yards. Find diagonal distance. **Formula:** $$d = \sqrt{80^2 + 115^2}$$ Calculate: $$d = \sqrt{6400 + 13225} = \sqrt{19625}$$ Approximate: $$d \approx 140.1$$ yards (rounded to nearest tenth) 8. **Problem:** Right triangle legs 2 and 4. Find hypotenuse length in simplest radical form. **Formula:** $$c = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20}$$ Simplify: $$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$$ **Answer:** $2\sqrt{5}$