Which option is correct?

If $xyz=-1$ , then the value of $\sqrt [ 4 ]{ \left( xyz \right)^4 }$ is

Option 1 : 1

Why?

Because

$\Large{ \sqrt [ 4 ]{ \left( xyz \right)^4} =\sqrt [ 4 ]{ \left( -1 \right) ^{ 4 } } =\sqrt [ 4 ]{ 1 } =1}$

Option 2 : -1

Why?

Because

$\Large{\sqrt [ 4 ]{ \left( xyz \right) ^{ 4 } }=xyz=-1}$

Option 3 : $i$

Why?

Because

$\Large{\sqrt [ 4 ]{ \left( xyz \right) ^{ 4 } } =\quad \sqrt [ 4 ]{ \left( xyz \right) ^{ 3 } } \times \sqrt [ 4 ]{ xyz } =\sqrt { i } \times \sqrt { i } =i}$

Explanation required

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

I think this one example may help you.
Ex. $(\sqrt{-1})^2 \neq \sqrt{(-1)^2}$
LHS : $(\sqrt{-1})^2 = -1$
RHS : $\sqrt{(-1)^2} = |-1| = 1$

I think all the options are correct. Given equation has 4 roots.i.e $1 , -1 , \iota , -\iota$

Akhil Bansal - 5 years, 5 months ago

Option $0!$

Hjalmar Orellana Soto - 5 years, 5 months ago

Option 1, because the property that (a)^(mn)=a^m*a^n is only preserved for real numbers.

Aditya Agarwal - 5 years, 5 months ago

Option 1. First condition: xyz= -1

ma pm - 5 years, 5 months ago

let x = xyz^4 so x will have roots ie i,-1,1,one more

genius literally - 5 years, 5 months ago

Option 1

Deepak Kumar - 5 years, 5 months ago

Explanation required

Department 8 - 5 years, 5 months ago

(x^2n)^(1/2n)=|x|

Deepak Kumar - 5 years, 5 months ago

@Deepak Kumar – Nice

Department 8 - 5 years, 5 months ago

Ya correct explanation.....its option 1

Ravi Dwivedi - 5 years, 5 months ago

option 1 guaranteed .option 2-no because the root sign stands for only positive values and option 3 -no because u cant do it for unreal numbers

Kaustubh Miglani - 5 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$