Question

Put the following radical expression into simplified form.

Answer 1

\[\sqrt[3]{\frac{ 1 }{ 7 }}\]
@whpalmer4

Answer 2

can you write that with fractional exponents?

Answer 3

I believe so, but don't know how.

Answer 4

\[\sqrt{x} = x^{1/2}\]\[\sqrt[3]{x} = x^{1/3}\]\[\sqrt[n]{x} = x^{1/n}\]

Answer 5

Ohhhhh, yes...I remember that now

Answer 6

also, do you remember the rule for a fraction under a radical?

\[\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\](that applies for any root, of course)

Answer 7

how to get rid of the radical on the bottom?

Answer 8

if you write it as fractional exponents, you could also rewrite the fraction as \[1^{1/3}*7^{-1/3}\]right? what is \(1^{1/3}=\)

Answer 9

uhh, 1?

Answer 10

yes. that's the primary cube root of 1. it turns out that there are two others, but we won't go into that right now.

so you could write that as \[\frac{1}{\sqrt[3]{7}}\]or\[7^{-1/3}\]which both seem simplified to me, but then again there's no formal definition of simplified form...

Some people don't like to have radical signs in the denominator, and insist that you "rationalize" the expression. Do you know how to do that with a square root? For example,
\[\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\]
after rationalization

Answer 11

multiply the top & bottom by the bottom
\[\frac{ 1 }{ \sqrt{2} }*\frac{ \sqrt{2} }{ \sqrt{2} }\]
So the bottom loses its radical

Answer 12

with cube roots it is a bit less intuitive: we have to multiply by the cube root twice to get rid of it

\[\frac{1}{\sqrt[3]{7}} = \frac{1}{\sqrt[3]{7}}*\frac{\sqrt[3]{7}*\sqrt[3]{7}}{\sqrt[3]{7}*\sqrt[3]{7}} = \frac{\sqrt[3]{7*7}}{7} = \frac{\sqrt[3]{49}}{7}\]
Does that seem simpler to you? :-)

Answer 13

@Jack1 , can you check this please?

Answer 14

if @whpalmer4 said it's right, then it's right, the guy's a master, hay

Answer 15

My online work doesn't agree, so I'll try the other way her put it

Answer 16

well if you continue with his working... hang on,

Answer 17

\[\huge \frac{ \sqrt[3]{49} }{ 7 } \]
u can rearrange this in other ways if it's looking for a particular way its written?
\[\huge \frac{ \sqrt[3]{7^2} }{ 7 } \]
or
\[\huge \frac{ 7^{\frac 23} }{ 7 } \]...?

Answer 18

I'll try the first one

Answer 19

As I probably said, there's no formal definition of what simplest means. I personally think that \(7^{-1/3}\) or \(\dfrac{1}{\sqrt[3]{7}}\) are both as simple as it will get, and rationalizing the denominator makes it less simple, but there are those who insist that denominators must be rationalized...

One thing I dislike about "modern" online education systems is that for ease of coding, they tend to be pretty inflexible about what they will accept as a correct answer, unlike a reasonable human grader. I don't have a problem with someone insisting on a specific answer format, so long as it is possible to learn what that answer format is without getting a bunch of problems with equivalent answers marked incorrect.

Answer 20

I know, i hate it sometimes too, ugh