Square Roots With Fractions?

Algebra Level 1

$\large \sqrt{ \frac xy} \times \sqrt{ \frac yz } \times \sqrt{ \frac zx }$

If $x,y,$ and $z$ are positive numbers, then what the value of the expression above?

The answer is 1.

10 solutions

Chew-Seong Cheong
Jun 22, 2015

$\sqrt{\dfrac{x}{y}} \times \sqrt{\dfrac{y}{z}} \times \sqrt{\dfrac{z}{x}} = \sqrt{\dfrac{x}{y} \times \dfrac{y}{z} \times \dfrac{z}{x}} = \sqrt{1} = \boxed{1}$

Mehul Arora
Jun 22, 2015

We know:- $\sqrt {\dfrac {a}{b}}= \dfrac {\sqrt {a}}{\sqrt {b}}$

$\Rightarrow \dfrac {\sqrt {x}}{\sqrt {y}} \times \dfrac {\sqrt {y}}{\sqrt {z}} \times \dfrac {\sqrt {x}}{\sqrt {z}}$

$\Huge \boxed {=1}$

easy pizy.

Mardokay Mosazghi - 5 years, 11 months ago

Not that easy.If $x=-1 \ , \ y=2 \ , \ z=-3$ , the answer is $(-1)$ .

Nihar Mahajan - 5 years, 11 months ago

you need to plug number and check, do the algebra

Mardokay Mosazghi - 5 years, 11 months ago

@Mardokay Mosazghi – It must be mentioned that $x,y,z$ are all either positive or negative. Otherwise the answer is $1$ or $-1$ .

Nihar Mahajan - 5 years, 11 months ago

You are right Nihar. Also, what if $x$ or $y$ or $z = 0$ ?

Arulx Z - 5 years, 11 months ago

@Arulx Z – The question has been edited and now $x,y,z$ are restricted to only positive reals.

Nihar Mahajan - 5 years, 11 months ago

Do read the N.B. ?? Please read the N.B.

Vubon Roy - 5 years, 11 months ago

@Vubon Roy – That NB was put 2 days after I commented this.

Nihar Mahajan - 5 years, 11 months ago

A Former Brilliant Member
Oct 28, 2015

The variables are understood to equal to 1

Aaron Tilton
Jun 24, 2015

x=4 y=2 z=1 becomes 2 x 2 x .25 = 1

Moderator note:

You have only shown that the expression equals to 1 when $x=4,y=2,z=1$ . But you should in fact show that the expression equals to 1 for all positive numbers $x,y$ and $z$ .