+0

# help

0
447
1

Let a^2=\frac{16}{44}$and$b^2=\frac{(2+\sqrt{5})^2}{11}$, where$a$is a negative real number and$b$is a positive real number. If (a+b)^3 can be expressed in the simplified form$\frac{x\sqrt{y}}{z}$where$x$,$y$, and$z$are positive integers, what is the value of the sum$x+y+z\$?

May 22, 2019

#1
+8966
+3

Let  $$a^2=\frac{16}{44}$$  and  $$b^2=\frac{(2+\sqrt{5})^2}{11}$$ , where  $$a$$  is a negative real number and  $$b$$  is a positive real number.

If  $$(a+b)^3$$  can be expressed in the simplified form  $$\frac{x\sqrt{y}}{z}$$  where  $$x$$,  $$y$$,  and  $$z$$  are positive integers,

what is the value of the sum  $$\vphantom{\frac{x\sqrt{y}}{z}}x+y+z$$ ?

______________________________________

$$a\quad=\quad-\sqrt{\frac{16}{44}}\quad=\quad -\frac{4}{2\sqrt{11}}\quad=\quad\frac{-2}{\sqrt{11}}\\~\\ b\quad=\quad\sqrt{\frac{(2+\sqrt{5})^2}{11}}\quad=\quad\frac{2+\sqrt{5}}{\sqrt{11}}\\~\\~\\ (a+b)^3\,=\,\Big(\frac{-2}{\sqrt{11}}+\frac{2+\sqrt{5}}{\sqrt{11}}\Big)^3\\~\\ (a+b)^3\,=\,\Big(\frac{-2+2+\sqrt{5}}{\sqrt{11}}\Big)^3\\~\\ (a+b)^3\,=\,\Big(\frac{\sqrt{5}}{\sqrt{11}}\Big)^3\\~\\ (a+b)^3\,=\, \frac{\sqrt{5}}{\sqrt{11}} \cdot\frac{\sqrt{5}}{\sqrt{11}}\cdot\frac{\sqrt{5}}{\sqrt{11}} \\~\\ (a+b)^3\,=\, \frac{5\sqrt{5}}{11\sqrt{11}} \\~\\ (a+b)^3\,=\, \frac{5\sqrt{5}}{11\sqrt{11}}\cdot\frac{\sqrt{11}}{\sqrt{11}} \\~\\ (a+b)^3\,=\, \frac{5\sqrt{55}}{121}$$

Now it is in the form  $$\frac{x\sqrt{y}}{z}$$  where  x,  y,  and  z  are positive integers.

x + y + z  =  5 + 55 + 121  =  181

May 22, 2019

#1
+8966
+3

Let  $$a^2=\frac{16}{44}$$  and  $$b^2=\frac{(2+\sqrt{5})^2}{11}$$ , where  $$a$$  is a negative real number and  $$b$$  is a positive real number.

If  $$(a+b)^3$$  can be expressed in the simplified form  $$\frac{x\sqrt{y}}{z}$$  where  $$x$$,  $$y$$,  and  $$z$$  are positive integers,

what is the value of the sum  $$\vphantom{\frac{x\sqrt{y}}{z}}x+y+z$$ ?

______________________________________

$$a\quad=\quad-\sqrt{\frac{16}{44}}\quad=\quad -\frac{4}{2\sqrt{11}}\quad=\quad\frac{-2}{\sqrt{11}}\\~\\ b\quad=\quad\sqrt{\frac{(2+\sqrt{5})^2}{11}}\quad=\quad\frac{2+\sqrt{5}}{\sqrt{11}}\\~\\~\\ (a+b)^3\,=\,\Big(\frac{-2}{\sqrt{11}}+\frac{2+\sqrt{5}}{\sqrt{11}}\Big)^3\\~\\ (a+b)^3\,=\,\Big(\frac{-2+2+\sqrt{5}}{\sqrt{11}}\Big)^3\\~\\ (a+b)^3\,=\,\Big(\frac{\sqrt{5}}{\sqrt{11}}\Big)^3\\~\\ (a+b)^3\,=\, \frac{\sqrt{5}}{\sqrt{11}} \cdot\frac{\sqrt{5}}{\sqrt{11}}\cdot\frac{\sqrt{5}}{\sqrt{11}} \\~\\ (a+b)^3\,=\, \frac{5\sqrt{5}}{11\sqrt{11}} \\~\\ (a+b)^3\,=\, \frac{5\sqrt{5}}{11\sqrt{11}}\cdot\frac{\sqrt{11}}{\sqrt{11}} \\~\\ (a+b)^3\,=\, \frac{5\sqrt{55}}{121}$$

Now it is in the form  $$\frac{x\sqrt{y}}{z}$$  where  x,  y,  and  z  are positive integers.

x + y + z  =  5 + 55 + 121  =  181

hectictar May 22, 2019