Answer:
Step-by-step explanation:
14.
A.
x = 5.0081818181..
10x = 50.08181818181....
100x = 500.81818181818...
100x - x = 500.8181818... - 5. 008181818....
99x = 495.81
[tex]x = \frac{495.81}{99} = \frac{495.81 \times 100}{99 \times 100} \\\\\\x= \frac{49581}{9900}[/tex]
[tex]x = \frac{5509}{1100}[/tex]
14.
B.
[tex]\sqrt{50} - \sqrt{98} + \sqrt{162} \\\\\sqrt{ 2 \times 25} - \sqrt{ 2 \times 49} + \sqrt{2 \times 81}\\\\\sqrt{ 2 \times 5^2} - \sqrt{ 2 \times 7^2} + \sqrt{2 \times 9^2}\\\\5\sqrt2 - 7 \sqrt 2 - 9 \sqrt 2\\\\\sqrt{2} ( 5 - 7 + 9 ) \\\\7\sqrt2[/tex]
15.
[tex]\frac{3 + \sqrt5}{4 - 2\sqrt 5} \\\\= \frac{3 + \sqrt5}{4 - 2\sqrt 5} \times \frac{4 + 2\sqrt5}{4 + 2\sqrt5}\\\\[/tex] [tex][ \ rationalizing \ the \ denominator \ ][/tex]
[tex]=\frac{(3 + \sqrt5)(4 + 2\sqrt5)}{(4 -2 \sqrt5)(4 + 2\sqrt5)}\\\\=\frac{12 + 6 \sqrt5 + 4\sqrt5 + 2(\sqrt5)^2}{(4)^2 - (2\sqrt5)^2}[/tex] [tex][ \ (a - b)(a +b) = a^2 - b^2 \ ][/tex]
[tex]=\frac{12 + 10 + 10\sqrt5}{16 - (4 \times 5)}\\\\=\frac{22 + 10\sqrt5}{16 - 20 }\\\\=\frac{22+ 10 \sqrt5 }{-4}[/tex]
[tex]= \frac{11}{-2} + \frac{5 \sqrt5}{-2}\\\\= -\frac{11}{2} + \frac{-5 }{2} \sqrt5\\\\[/tex]
[tex]a = - \frac{11}{2} , \ b = -\frac{5}{2}[/tex]
16
[tex]\frac{\sqrt7 + 1 }{\sqrt7 - 1} - \frac{\sqrt7 - 1}{\sqrt7 + 1}\\\\[/tex]
[tex]\frac{(\sqrt 7 + 1 )( \sqrt7 + 1)}{(\sqrt7 - 1 )(\sqrt7 + 1)} - \frac{(\sqrt7 - 1)(\sqrt 7 - 1)}{(\sqrt7 + 1)(\sqrt7 - 1)}[/tex] [tex][ \ LCM \ of \ denominators : (\sqrt7 -1)(\sqrt7 + 1) \ ][/tex]
[tex]\frac{(\sqrt 7 + 1 )^2 -( \sqrt7 - 1)^2}{(\sqrt7)^2 - 1^2}[/tex] [tex][ \ (a \pm b)(a \pm b) = (a \pm b)^2 \ (a-b)(a+b) = a^2 - b^2 \ ][/tex]
[tex]=\frac{(7 + 1 + 2\sqrt7) - ( 7 + 1 - 2\sqrt7)}{7 - 1}\\\\=\frac{8 + 2\sqrt7 - 8 + 2\sqrt7}{6}\\\\= \frac{4 \sqrt7}{6}\\\\= \frac{2 \sqrt 7}{3}[/tex]
