2 Problems involving radicals.

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1) Simplify \(\displaystyle \L \sqrt[4]{x^2}\, +\, \sqrt[4]{x^6}\)

I don't know if it's right but is it...:

. . .\(\displaystyle \L x^{\frac{1}{2}}\, +\, x^{\frac{3}{2}}\)

. . .\(\displaystyle \L \sqrt[2]{x}\, +\, \sqrt[2]{x^3}\)

. . .\(\displaystyle \L \sqrt{x}\, +\, x\,\sqrt[2]{x}\)

. . .\(\displaystyle \L 1\, +\, x\,\sqrt[2]{x}\)

2) Solve the following for \(\displaystyle d\):

. . .\(\displaystyle \L r\, =\, \sqrt[3]{\frac{3w}{4\pi d}}\)

No clue except cubing both sides to get rid of the radical.

Thanks.
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Edited by stapel -- Reason for edit: replacing bitmaps with LaTeX
 
Hello, AirForceOne!

\(\displaystyle \text{Simplify: }\,\sqrt[4]{x^2} \,+ \,\sqrt[4]{x^6}\)

You were okay up to the last step . . .

We have: \(\displaystyle \L\,x^{\frac{2}{4}} \,+\,x^{\frac{6}{4}}\;=\;x^{\frac{1}{2}}\,+\,x^{\frac{3}{2}} \;=\;\sqrt{x}\,+\,x\sqrt{x}\) . . . so you were correct.

But it cannot be simplied . . . just factored: \(\displaystyle \L\:\sqrt{x}\left(1\,+\,x\right)\)



\(\displaystyle \text{Solve for }d:\L\;\;r\:=\:\sqrt[3]{\frac{3w}{4\pi d}}\)

You're right . . . cube both sides: \(\displaystyle \L\: r^3\;=\;\frac{3w}{4\pi d}\)

Multiply by \(\displaystyle 4\pi d:\L\;\;4\pi dr^3 \;= \;3w\)

Divide by \(\displaystyle 4\pi r^3:\L\;\;\fbox{d \;= \;\frac{3w}{4\pi r^3}}\)

 
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