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### Simplifying by Rationalizing the Denominator
(a) [tex]\(\frac{1+\sqrt{7}}{1-\sqrt{7}}\)[/tex]
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, [tex]\(1 + \sqrt{7}\)[/tex]:
[tex]\[ \frac{(1 + \sqrt{7})(1 + \sqrt{7})}{(1 - \sqrt{7})(1 + \sqrt{7})} \][/tex]
Expanding both the numerator and the denominator:
Numerator:
[tex]\[ (1 + \sqrt{7})(1 + \sqrt{7}) = 1 + 2\sqrt{7} + 7 = 8 + 2\sqrt{7} \][/tex]
Denominator:
[tex]\[ (1 - \sqrt{7})(1 + \sqrt{7}) = 1 - 7 = -6 \][/tex]
Thus:
[tex]\[ \frac{8 + 2\sqrt{7}}{-6} = -\frac{8}{6} - \frac{2\sqrt{7}}{6} = -\frac{4}{3} - \frac{\sqrt{7}}{3} \][/tex]
The simplified result is approximately [tex]\(-2.21525043702153\)[/tex].
---
(b) [tex]\(\frac{1}{\sqrt{6}-\sqrt{5}}\)[/tex]
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, [tex]\(\sqrt{6} + \sqrt{5}\)[/tex]:
[tex]\[ \frac{1 \cdot (\sqrt{6} + \sqrt{5})}{(\sqrt{6} - \sqrt{5})(\sqrt{6} + \sqrt{5})} \][/tex]
Expanding both the numerator and the denominator:
Numerator:
[tex]\[ \sqrt{6} + \sqrt{5} \][/tex]
Denominator:
[tex]\[ (\sqrt{6} - \sqrt{5})(\sqrt{6} + \sqrt{5}) = 6 - 5 = 1 \][/tex]
Thus:
[tex]\[ \sqrt{6} + \sqrt{5} \][/tex]
The simplified result is approximately [tex]\(4.685557720282975\)[/tex].
---
(c) [tex]\(\frac{1}{2 \sqrt{5}-\sqrt{3}}\)[/tex]
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, [tex]\(2 \sqrt{5} + \sqrt{3}\)[/tex]:
[tex]\[ \frac{1 \cdot (2 \sqrt{5} + \sqrt{3})}{(2 \sqrt{5} - \sqrt{3})(2 \sqrt{5} + \sqrt{3})} \][/tex]
Expanding both the numerator and the denominator:
Numerator:
[tex]\[ 2 \sqrt{5} + \sqrt{3} \][/tex]
Denominator:
[tex]\[ (2 \sqrt{5})^2 - (\sqrt{3})^2 = 20 - 3 = 17 \][/tex]
Thus:
[tex]\[ \frac{2 \sqrt{5} + \sqrt{3}}{17} \][/tex]
The simplified result is approximately [tex]\(0.36495216250402684\)[/tex].
---
(d) [tex]\(\frac{5+2 \sqrt{3}}{7+4 \sqrt{3}}\)[/tex]
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, [tex]\(7 - 4 \sqrt{3}\)[/tex]:
[tex]\[ \frac{(5+2 \sqrt{3})(7 - 4 \sqrt{3})}{(7+4 \sqrt{3})(7-4 \sqrt{3})} \][/tex]
Expanding both the numerator and the denominator:
Numerator:
[tex]\[ (5+2 \sqrt{3})(7 - 4 \sqrt{3}) = 35 - 20 \sqrt{3} + 14 \sqrt{3} - 24 = 11 - 6 \sqrt{3} \][/tex]
Denominator:
[tex]\[ (7+4 \sqrt{3})(7-4 \sqrt{3}) = 49 - (4 \sqrt{3})^2 = 49 - 48 = 1 \][/tex]
Thus:
[tex]\[ 11 - 6 \sqrt{3} \][/tex]
The simplified result is approximately [tex]\(0.6076951545867362\)[/tex].
---
(e) [tex]\(\frac{4 \sqrt{3}+5 \sqrt{2}}{\sqrt{48}+\sqrt{18}}\)[/tex]
First simplify the denominator:
[tex]\[ \sqrt{48} = 4 \sqrt{3}, \quad \sqrt{18} = 3 \sqrt{2} \][/tex]
Thus the expression becomes:
[tex]\[ \frac{4 \sqrt{3} + 5 \sqrt{2}}{4 \sqrt{3} + 3 \sqrt{2}} \][/tex]
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, [tex]\(4 \sqrt{3} - 3 \sqrt{2}\)[/tex]:
[tex]\[ \frac{(4 \sqrt{3} + 5 \sqrt{2})(4 \sqrt{3} - 3 \sqrt{2})}{(4 \sqrt{3} + 3 \sqrt{2})(4 \sqrt{3} - 3 \sqrt{2})} \][/tex]
Expanding both the numerator and the denominator:
Numerator:
[tex]\[ (4 \sqrt{3} + 5 \sqrt{2})(4 \sqrt{3} - 3 \sqrt{2}) = 48 - 15 \sqrt{6} \][/tex]
Denominator:
[tex]\[ (4 \sqrt{3})^2 - (3 \sqrt{2})^2 = 48 - 18 = 30 \][/tex]
Thus:
[tex]\[ \frac{48 - 15 \sqrt{6}}{30} = \frac{48}{30} - \frac{15 \sqrt{6}}{30} = \frac{8}{5} - \frac{\sqrt{6}}{2} \][/tex]
The simplified result is approximately [tex]\(1.2531972647421807\)[/tex].
(a) [tex]\(\frac{1+\sqrt{7}}{1-\sqrt{7}}\)[/tex]
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, [tex]\(1 + \sqrt{7}\)[/tex]:
[tex]\[ \frac{(1 + \sqrt{7})(1 + \sqrt{7})}{(1 - \sqrt{7})(1 + \sqrt{7})} \][/tex]
Expanding both the numerator and the denominator:
Numerator:
[tex]\[ (1 + \sqrt{7})(1 + \sqrt{7}) = 1 + 2\sqrt{7} + 7 = 8 + 2\sqrt{7} \][/tex]
Denominator:
[tex]\[ (1 - \sqrt{7})(1 + \sqrt{7}) = 1 - 7 = -6 \][/tex]
Thus:
[tex]\[ \frac{8 + 2\sqrt{7}}{-6} = -\frac{8}{6} - \frac{2\sqrt{7}}{6} = -\frac{4}{3} - \frac{\sqrt{7}}{3} \][/tex]
The simplified result is approximately [tex]\(-2.21525043702153\)[/tex].
---
(b) [tex]\(\frac{1}{\sqrt{6}-\sqrt{5}}\)[/tex]
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, [tex]\(\sqrt{6} + \sqrt{5}\)[/tex]:
[tex]\[ \frac{1 \cdot (\sqrt{6} + \sqrt{5})}{(\sqrt{6} - \sqrt{5})(\sqrt{6} + \sqrt{5})} \][/tex]
Expanding both the numerator and the denominator:
Numerator:
[tex]\[ \sqrt{6} + \sqrt{5} \][/tex]
Denominator:
[tex]\[ (\sqrt{6} - \sqrt{5})(\sqrt{6} + \sqrt{5}) = 6 - 5 = 1 \][/tex]
Thus:
[tex]\[ \sqrt{6} + \sqrt{5} \][/tex]
The simplified result is approximately [tex]\(4.685557720282975\)[/tex].
---
(c) [tex]\(\frac{1}{2 \sqrt{5}-\sqrt{3}}\)[/tex]
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, [tex]\(2 \sqrt{5} + \sqrt{3}\)[/tex]:
[tex]\[ \frac{1 \cdot (2 \sqrt{5} + \sqrt{3})}{(2 \sqrt{5} - \sqrt{3})(2 \sqrt{5} + \sqrt{3})} \][/tex]
Expanding both the numerator and the denominator:
Numerator:
[tex]\[ 2 \sqrt{5} + \sqrt{3} \][/tex]
Denominator:
[tex]\[ (2 \sqrt{5})^2 - (\sqrt{3})^2 = 20 - 3 = 17 \][/tex]
Thus:
[tex]\[ \frac{2 \sqrt{5} + \sqrt{3}}{17} \][/tex]
The simplified result is approximately [tex]\(0.36495216250402684\)[/tex].
---
(d) [tex]\(\frac{5+2 \sqrt{3}}{7+4 \sqrt{3}}\)[/tex]
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, [tex]\(7 - 4 \sqrt{3}\)[/tex]:
[tex]\[ \frac{(5+2 \sqrt{3})(7 - 4 \sqrt{3})}{(7+4 \sqrt{3})(7-4 \sqrt{3})} \][/tex]
Expanding both the numerator and the denominator:
Numerator:
[tex]\[ (5+2 \sqrt{3})(7 - 4 \sqrt{3}) = 35 - 20 \sqrt{3} + 14 \sqrt{3} - 24 = 11 - 6 \sqrt{3} \][/tex]
Denominator:
[tex]\[ (7+4 \sqrt{3})(7-4 \sqrt{3}) = 49 - (4 \sqrt{3})^2 = 49 - 48 = 1 \][/tex]
Thus:
[tex]\[ 11 - 6 \sqrt{3} \][/tex]
The simplified result is approximately [tex]\(0.6076951545867362\)[/tex].
---
(e) [tex]\(\frac{4 \sqrt{3}+5 \sqrt{2}}{\sqrt{48}+\sqrt{18}}\)[/tex]
First simplify the denominator:
[tex]\[ \sqrt{48} = 4 \sqrt{3}, \quad \sqrt{18} = 3 \sqrt{2} \][/tex]
Thus the expression becomes:
[tex]\[ \frac{4 \sqrt{3} + 5 \sqrt{2}}{4 \sqrt{3} + 3 \sqrt{2}} \][/tex]
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, [tex]\(4 \sqrt{3} - 3 \sqrt{2}\)[/tex]:
[tex]\[ \frac{(4 \sqrt{3} + 5 \sqrt{2})(4 \sqrt{3} - 3 \sqrt{2})}{(4 \sqrt{3} + 3 \sqrt{2})(4 \sqrt{3} - 3 \sqrt{2})} \][/tex]
Expanding both the numerator and the denominator:
Numerator:
[tex]\[ (4 \sqrt{3} + 5 \sqrt{2})(4 \sqrt{3} - 3 \sqrt{2}) = 48 - 15 \sqrt{6} \][/tex]
Denominator:
[tex]\[ (4 \sqrt{3})^2 - (3 \sqrt{2})^2 = 48 - 18 = 30 \][/tex]
Thus:
[tex]\[ \frac{48 - 15 \sqrt{6}}{30} = \frac{48}{30} - \frac{15 \sqrt{6}}{30} = \frac{8}{5} - \frac{\sqrt{6}}{2} \][/tex]
The simplified result is approximately [tex]\(1.2531972647421807\)[/tex].
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