MHB Simplify expression with exponents

[ahmedb]

simplify and answer should be in positive exponents.
(((4x^6)^3(4y^-8))/((2x)^4(12y^3)^2))^1/2
please help and thanks

 

[Also sprach Zarathustra]

Re: simplify

simplify and answer should be in positive exponents.
(((4x^6)^3(4y^-8))/((2x)^4(12y^3)^2))^1/2
please help and thanks

$$ \huge{(}\frac{{(4x^6)^3}4y^{-8}}{(2x)^4(12y^3)^2}\huge{)}^{\frac{1}{2}} $$

$$ \huge{(}\frac{{(4^3x^{18})}4y^{-8}}{(2^4x^4)(12^2y^6)}\huge{)}^{\frac{1}{2}} $$

$$ \huge{(}\frac{{(64x^{18})}4y^{-8}}{(16x^4)(144y^6)}\huge{)}^{\frac{1}{2}} $$

$$ \huge{(}\frac{4x^{14}}{36y^{14}}\huge{)}^{\frac{1}{2}} $$

$$ \huge{(}\frac{x^{14}}{9y^{14}}\huge{)}^{\frac{1}{2}} $$

$$ \huge{(}(\frac{x}{9y})^{14}\huge{)}^{\frac{1}{2}} $$

$$ (\frac{x}{9y})^{7} $$

 

[battleman13]

Re: simplify

You should probably show any work you have tried first so that more importantly we can fix any misconceptions you may have about this process.

If your going any further in math the ability to do the work in this problem will be required.

You may now have the answer, but what you really need is the ability to reach it on your own.

 

[soroban]

Hello, MoneyKing!

$\text{Simplify: }\:\left[\dfrac{(4x^6)^3(4y^{-8})}{(2x)^4(12y^3)^2}\right]^{\frac{1}{2}}$

$\left[\dfrac{(4x^6)^3(4y^{-8})}{(2x)^4(12y^3)^2}\right]^{\frac{1}{2}} \;=\;\;\left[\dfrac{4^3(x^6)^3\cdot 4y^{-8}}{2^4x^4\cdot 12^2(y^3)^2}\right]^{\frac{1}{2}} \;=\;\;\left[\dfrac{64x^{18}\cdot 4y^{-8}}{16x^4\cdot144y^6}\right]^{\frac{1}{2}} $

. . . . . $=\;\;\left[\dfrac{x^{14}}{9y^{14}}\right]^{\frac{1}{2}} \;=\;\;
\dfrac{(x^{14})^{\frac{1}{2}}}{9^{\frac{1}{2}}(y^{14})^{\frac{1}{2}}} \;=\;\;\dfrac{x^7}{3y^7} $