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1. AngelaB97

|dw:1441684069299:dw|

2. AngelaB97

how would you solve this?

3. AngelaB97

@Jhannybean

4. Jhannybean

$\large \sqrt[5]{\frac{1}{8}} \cdot \sqrt[5]{\frac{1}{4}}$

5. Jhannybean

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?

6. AngelaB97

(1/4) ^1/5

7. Jhannybean

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 $\left(\frac{1}{8^{1/5}}\right)\left(\frac{1}{4^{1/5}}\right)$

8. Jhannybean

And now we just multiply.

9. Jhannybean

$\frac{1}{8^{1/5} \cdot 4^{1/5}} = \frac{1}{32^{1/5}}$

10. Jhannybean

Now an easy way to think about it is finding the prime factors of 32. What are they?

11. AngelaB97

8 and 4?

12. Jhannybean

Yes, and if we break 8 and 4 down even further, what would we get?

13. AngelaB97

|dw:1441685366603:dw|

14. Jhannybean

Is that a question mark?

15. Jhannybean

|dw:1441685422951:dw|

16. AngelaB97

question mark lol

17. Jhannybean

Ok haha.

18. Jhannybean

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.

19. Jhannybean

Does that make sense, @AngelaB97 ?

20. Jhannybean

Therefore $$\dfrac{1}{32^{1/5}} = \boxed{\dfrac{1}{2}}$$

21. AngelaB97

yes thanks so much once again @Jhannybean

22. AngelaB97

sorry to bother you again

23. Jhannybean

No problem :)

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