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Simplify $ \root 3 \of {x \root 3 \of {x \root 3 \of {x \sqrt{x}}}}. $ Express your answer in simplest radical form in terms of $x$.

AWESOMEEE  Jun 30, 2015

Best Answer 

 #2
avatar+18712 
+5

Simplify $ \root 3 \of {x \root 3 \of {x \root 3 \of {x \sqrt{x}}}}. $ Express your answer in simplest radical form in terms of $x$.

 

$$\boxed{~~ \text{Formula: } \sqrt[n]{a\cdot b}
= \sqrt[n]{a}\cdot \sqrt[n]{b} ~~}$$

 

$$\root 3 \of {x \root 3 \of {x \root 3 \of {x \sqrt{x}}}} \\\\
= \root 3 \of {x} \cdot \root 3 \of {\root 3 \of {x \root 3 \of {x \sqrt{x}}}}\\\\
=
\root 3 \of {x} \cdot
\root 9 \of {x \root 3 \of {x \sqrt{x}}}\\\\
=
\root 3 \of {x} \cdot
\root 9 \of {x} \cdot
\root 9 \of {\root 3 \of {x \sqrt{x}}}\\\\
=
\root 3 \of {x} \cdot
\root 9 \of {x} \cdot
\root 27 \of {x \sqrt{x}}\\\\
=
\root 3 \of {x} \cdot
\root 9 \of {x} \cdot
\root 27 \of {x} \cdot
\root 27 \of {\sqrt{x}}\\\\
=
\root 3 \of {x} \cdot
\root 9 \of {x} \cdot
\root 27 \of {x} \cdot
\root 54 \of {x}\\\\
=
x^{\frac13}
\cdot x^{\frac19}
\cdot x^{\frac1{27}}
\cdot x^{\frac1{54}}\\\\
=
x^{\frac13+\frac19+\frac1{27}+ \frac1{54}}\\\\
=
x^{\frac13 \cdot \frac{18}{18}
+\frac19 \cdot \frac{6}{6}
+\frac1{27} \cdot \frac{2}{2} + \frac1{54}}\\\\
=x^{\frac{18+6+2+1}{54} } \\\\$$

 

$$\\=x^{\frac{27}{54}}\\\\
=x^{\frac{1}{2} } \\\\
\mathbf{= \sqrt{x}}$$

 

heureka  Jul 1, 2015
Sort: 

3+0 Answers

 #1
avatar+78557 
+5

√x

 

Basically, we're taking the cube root of x^(3/2) = x^(1/2) and multiplying this by x  = (x)^(3/2)....then taking the cube root of that, etc.........the last operation results in  (x)^(3/2)^(1/3)  = x^(1/2)  =

√x

 

 

CPhill  Jun 30, 2015
 #2
avatar+18712 
+5
Best Answer

Simplify $ \root 3 \of {x \root 3 \of {x \root 3 \of {x \sqrt{x}}}}. $ Express your answer in simplest radical form in terms of $x$.

 

$$\boxed{~~ \text{Formula: } \sqrt[n]{a\cdot b}
= \sqrt[n]{a}\cdot \sqrt[n]{b} ~~}$$

 

$$\root 3 \of {x \root 3 \of {x \root 3 \of {x \sqrt{x}}}} \\\\
= \root 3 \of {x} \cdot \root 3 \of {\root 3 \of {x \root 3 \of {x \sqrt{x}}}}\\\\
=
\root 3 \of {x} \cdot
\root 9 \of {x \root 3 \of {x \sqrt{x}}}\\\\
=
\root 3 \of {x} \cdot
\root 9 \of {x} \cdot
\root 9 \of {\root 3 \of {x \sqrt{x}}}\\\\
=
\root 3 \of {x} \cdot
\root 9 \of {x} \cdot
\root 27 \of {x \sqrt{x}}\\\\
=
\root 3 \of {x} \cdot
\root 9 \of {x} \cdot
\root 27 \of {x} \cdot
\root 27 \of {\sqrt{x}}\\\\
=
\root 3 \of {x} \cdot
\root 9 \of {x} \cdot
\root 27 \of {x} \cdot
\root 54 \of {x}\\\\
=
x^{\frac13}
\cdot x^{\frac19}
\cdot x^{\frac1{27}}
\cdot x^{\frac1{54}}\\\\
=
x^{\frac13+\frac19+\frac1{27}+ \frac1{54}}\\\\
=
x^{\frac13 \cdot \frac{18}{18}
+\frac19 \cdot \frac{6}{6}
+\frac1{27} \cdot \frac{2}{2} + \frac1{54}}\\\\
=x^{\frac{18+6+2+1}{54} } \\\\$$

 

$$\\=x^{\frac{27}{54}}\\\\
=x^{\frac{1}{2} } \\\\
\mathbf{= \sqrt{x}}$$

 

heureka  Jul 1, 2015
 #3
avatar+90970 
0

You made that look REALLY hard Heureka   LOL   :))

Melody  Jul 2, 2015

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