How do I simplify cubed roots with multiple variables and coefficients?

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The discussion focuses on simplifying the expression ³√9xy^4 * ³√12x³y. The user attempts to break down the cubed roots but struggles with the simplification of 9 under the radical. A key suggestion is to combine all components under a single cubic root for easier manipulation. The final simplified answer is 3xy * ³√4xy², indicating that coefficients and variables can be rearranged for simplification. The conversation highlights the importance of understanding how to manage multiple variables and coefficients in radical expressions.
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Homework Statement


Simplify...

Homework Equations


Starting Equation:

³√9xy^4 * ³√12x³y

The Attempt at a Solution



(Sorry ahead of time if this is sloppy)

³√9xy^4 * ³√12x³y - I broke up each part of the cubed roots to try and simplify them³√9 * ³√x * ³√y * ³√y³ * ³√y * ³√12 * ³√x³ * ³√y³ - I then simplified the obvious cubed roots, and simplified √12 to 2√3:

y * ³√9xy * 2x * ³√3y - This is where I'm stuck.I don't understand how I can simplify if further when 9 has no rational number that can be multiplied by itself 3 times to get to 9, yet apparently the answer to it doesn't have 9 under the radical.

The answer, to help out anybody who wants to help me, is:

3xy * ³√4xy²Thanks for any help.
 
Last edited:
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What is really under the roots? Use parentheses! Bring everything under a single cubic root. ehild
 
ehild said:
What is really under the roots? Use parentheses! Bring everything under a single cubic root. ehild
Alright, thanks for the help. So, since I have a coefficient and similar variables under the cubed roots, I can change around where they all go?

And at first, I thought ehild was trying to call me a child for not knowing the answer, and just misspelled it...
 
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Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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