can someone please help me understand radicals??

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can someone please help me understand radicals??

Mathematics
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|dw:1441684069299:dw|
how would you solve this?

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\[\large \sqrt[5]{\frac{1}{8}} \cdot \sqrt[5]{\frac{1}{4}}\]
Let's start by rewriting this as a fraction with a power. Recall that \(\large \sqrt[n]{x^m} =(x^m)^n\)\[\left(\frac{1}{8}\right)^{1/5}\]Can you tellme how we would write the other fraction?
(1/4) ^1/5
Awesome. Now we multiply them together. \[\left(\frac{1}{8}\right)^{1/5}\cdot \left(\frac{1}{4}\right)^{1/5}\] Next we distribute the fractional power to all terms within the parenthesis \[\left(\frac{1}{8^{1/5}}\right)\left(\frac{1}{4^{1/5}}\right)\]
And now we just multiply.
\[\frac{1}{8^{1/5} \cdot 4^{1/5}} = \frac{1}{32^{1/5}} \]
Now an easy way to think about it is finding the prime factors of 32. What are they?
8 and 4?
Yes, and if we break 8 and 4 down even further, what would we get?
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Is that a question mark?
|dw:1441685422951:dw|
question mark lol
Ok haha.
You're right about the 2, Now let's break 32 apart. \[(32)^{1/5} = (2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2)^{1/5}\] The \(\frac{1}{5}\) power represents the 5th root. This means that for every PAIR of FIVE numbers, 1 will come out of the 5th root. Therefore since there are exactly five 2's under the 5th root, we are going to only use one 2.
Does that make sense, @AngelaB97 ?
Therefore \(\dfrac{1}{32^{1/5}} = \boxed{\dfrac{1}{2}}\)
yes thanks so much once again @Jhannybean
sorry to bother you again
No problem :)

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