Rationalize and simplify as a mixed radical

[hervorragend]

\(\displaystyle \mbox{c) }\, \sqrt[4]{\strut \dfrac{162\, x^6}{y^4} \,}\)

How do I do *title* for this question ?

 

[stapel]

How do I do *title* for this question ?

You fill in the "Title" box (as you did with the text "Rationalize and simplify as a mixed radical").

\(\displaystyle \mbox{c) }\, \sqrt[4]{\strut \dfrac{162\, x^6}{y^4} \,}\)

I will guess that the subject line contains the instructions; namely, you are supposed to rationalize the denominator and simplify the expression in terms of radicals.

Where are you stuckin the process? You've factored the 162 into its primes, you've noted that the denominator rationalizes via simplification (the radical "cancelling" with the exponent), and... then what? Please be complete. Thank you!

 

[ksdhart]

I'd begin by rewriting the expression. Recall that the nth-root of a fraction is equal to the nth-root of the numerator divided by the nth root of the denominator. Like so:

\(\displaystyle \sqrt[4]{\frac{162x^6}{y^4}}=\frac{\sqrt[4]{162x^6}}{\sqrt[4]{y^4}}\)

Can you see how to proceed from here? If not, that's okay. But when you reply back, please show any steps you took so we can see where you are stuck.

 

[hervorragend]

Thanks for the replies guys I'm stuck with the x^6. What do I do with it?

what I've done so far

 

[ksdhart]

Looks good so far. There's not much more that can be done. I think the only simplification that can be made is:

\(\displaystyle \frac{3\sqrt[4]{2x^6}}{y}=\frac{3\sqrt[4]{2}\sqrt[4]{x^6}}{y}\)

Then you can simplify down the 4th root of x^6. Remember the exponent rules. The 4th root of x can be written as x^1/4. So how would you simplify (x⁶)^1/4?

 

[lookagain]

Looks good so far. There's not much more that can be done. I think the only simplification that can be made is:

\(\displaystyle \frac{3\sqrt[4]{2x^6}}{y}=\frac{3\sqrt[4]{2}\sqrt[4]{x^6}}{y} \ \ \ \ \ \) <------ You don't do this.

Then you can simplify down the 4th root of x^6. Remember the exponent rules. The 4th root of x can be written as x^1/4. So how would you simplify (x⁶)^1/4?

No, you leave the "2" inside as part of the radicand.


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hervorragend, you would continue with the equivalent of:


\(\displaystyle \dfrac{3\sqrt[4]{2x^6}}{y} \ = \ \)


\(\displaystyle \dfrac{3\sqrt[4]{(x^4)(2x^2)}}{y} \ = \)


\(\displaystyle \boxed{ \ \dfrac{3x\sqrt[4]{2x^2}}{y} \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \)This problem has an unstated assumption that x is a non-negative variable, and that y is a positive variable.