Math Help - Quickie #3

1. Quickie #3

Simplify: . $\frac{(4 + \sqrt{15})^{\frac{3}{2}} + (4 - \sqrt{15})^{\frac{3}{2}}} {(6 + \sqrt{35})^{\frac{3}{2}} - (6 - \sqrt{35})^{\frac{3}{2}}}$

2. Originally Posted by Soroban
Simplify: . $\frac{(4 + \sqrt{15})^{\frac{3}{2}} + (4 - \sqrt{15})^{\frac{3}{2}}} {(6 + \sqrt{35})^{\frac{3}{2}} - (6 - \sqrt{35})^{\frac{3}{2}}}$

] $\frac{(4 + \sqrt{15})^{\frac{3}{2}} + (4 - \sqrt{15})^{\frac{3}{2}}} {(6 + \sqrt{35})^{\frac{3}{2}} - (6 - \sqrt{35})^{\frac{3}{2}}}$

$(4+\sqrt{15})^{\frac{3}{2}}=\frac{7\sqrt{10}}{2}+\ frac{9\sqrt{6}}{2}$...[a]

$(4-\sqrt{15})^{\frac{3}{2}}=\frac{7\sqrt{10}}{2}-\frac{9\sqrt{6}}{2}$...[b]

$(6+\sqrt{35})^{\frac{3}{2}}=\frac{11\sqrt{14}}{2}+ \frac{13\sqrt{10}}{2}$....[c]

$(6-\sqrt{35})^{\frac{3}{2}}=\frac{11\sqrt{14}}{2}-\frac{13\sqrt{10}}{2}$...[d]

$\frac{a+b}{c-d}=\frac{7\sqrt{10}}{13\sqrt{10}}=\boxed{\frac{7}{ 13}}$

3. Originally Posted by Soroban
Simplify: . $\frac{(4 + \sqrt{15})^{\frac{3}{2}} + (4 - \sqrt{15})^{\frac{3}{2}}} {(6 + \sqrt{35})^{\frac{3}{2}} - (6 - \sqrt{35})^{\frac{3}{2}}}$

Hello Soroban,,
I really don't know why you call this problem a "Quickie" (I still have a crack in my head!)

It took some time until I remembered the property:

If x > y > 0 then

$\sqrt{x+y}=\sqrt{\frac{x+\sqrt{x^2-y^2}}{2}}+\sqrt{\frac{x-\sqrt{x^2-y^2}}{2}}$ or

$\sqrt{x-y}=\sqrt{\frac{x+\sqrt{x^2-y^2}}{2}}-\sqrt{\frac{x-\sqrt{x^2-y^2}}{2}}$

With these properties your term becomes:

$\frac{(4+\sqrt{15})^{\frac{3}{2}}+(4-\sqrt{15})^{\frac{3}{2}}} {(6+\sqrt{35})^{\frac{3}{2}}-(6-\sqrt{35})^{\frac{3}{2}}}$ = $\frac{(\sqrt{\frac{5}{2}}+\sqrt{\frac{3}{2}})^3+(\ sqrt{\frac{5}{2}}-\sqrt{\frac{3}{2}})^3}{(\sqrt{\frac{7}{2}}+\sqrt{\ frac{5}{2}})^3-(\sqrt{\frac{7}{2}}-\sqrt{\frac{5}{2}})^3}$

Expand the brackets and collect the roots with the same value. I've got:

$\frac{7 \cdot \sqrt{10}}{13 \cdot \sqrt{10}}=\frac{7}{13}$

Hapoooh!

EB

4. Lovely work, Galactus and EB!

Don't know if the "Quickie" solution is any faster . . .

We have: . $\frac{(4 + \sqrt{15})^{\frac{3}{2}} + (4 - \sqrt{15})^{\frac{3}{2}}} {(6 + \sqrt{35})^{\frac{3}{2}} - (6 - \sqrt{35})^{\frac{3}{2}}}$

Multiply top and bottom by $2^{\frac{3}{2}}\!:$

. . $\frac{(8 + 2\sqrt{15})^{\frac{3}{2}} + (8 - 2\sqrt{15})^{\frac{3}{2}}} {(12 + 2\sqrt{35})^{\frac{3}{2}} - (12 + 2\sqrt{35})^{\frac{3}{2}}}$

. . $=\;\frac{\left[(\sqrt{5}+\sqrt{3})^2\right]^{\frac{3}{2}} + \left[(\sqrt{5} - \sqrt{3})^2\right]^{\frac{3}{2}}} {\left[(\sqrt{7} + \sqrt{5})^2\right]^{\frac{3}{2}} - \left[(\sqrt{7} - \sqrt{5})^2\right]^{\frac{3}{2}} }$

. . $= \;\frac{(\sqrt{5}+\sqrt{3})^3 + (\sqrt{5} - \sqrt{3})^3}{(\sqrt{7} + \sqrt{5})^3 - (\sqrt{7} - \sqrt{5})^3}$

. . $=\:\frac{(5\sqrt{5} + 15\sqrt{3} + 9\sqrt{5} + 3\sqrt{3}) + (5\sqrt{5} - 15\sqrt{3} + 9\sqrt{5} - 3\sqrt{3})} {(7\sqrt{7} + 21\sqrt{5} + 15\sqrt{7} + 5\sqrt{5}) - (7\sqrt{7} - 21\sqrt{5} + 15\sqrt{7} - 5\sqrt{5}}$

. . $= \:\frac{28\sqrt{5}}{52\sqrt{5}}$

. . $=\:\boxed{\frac{7}{13}}$