Question

Simplify by reducing the index of the radical (Help? I don't understand how to do this!):

Answer 1

\[\sqrt[12]{125x ^{3}}\]

Answer 2

Hint:
What if we re-write it this way:

\(\sqrt[12]{5^3x^3}\)

Then change the radical to a fraction to get this:

\((5x)^{\frac{3}{12}}\)

Answer 3

That fraction can also be simplified. After simplification, you can change the fraction back to a radical.

Answer 4

So does that mean the radical would look like: \[\sqrt[4]{}\]

Answer 5

Oops, \[\sqrt[4]{5x}\]

Answer 6

Correct

Answer 7

So that's it? Can you help me on one more example so I can fully understand?

Answer 8

The only step you're probably confused on is this:

\(a^xb^x\) = \((ab)^x\)

Answer 9

what about this then? since it has two different variables and exponents: \[\sqrt[49]{y ^{21}z ^{14}}\]

Answer 10

@Hero

Answer 11

The variables don't really matter. If you notice, the example I showed you also has two variables.

Answer 12

If anything, that one is even easier since 21/49 = 3/7 and 14/49 = 2/7

Answer 13

So do you do each individually...like: \[y ^{21/49}\]

Answer 14

Yup

Answer 15

and \[z ^{14/49}\]

Answer 16

But after you simplify the fraction, put it back in root form

Answer 17

After you simplify those they go to \[\sqrt[7]{y ^{3}z ^{2}}\]

Answer 18

?

Answer 19

Looks like you have it all figured out.

Answer 20

Thanks so much!!

Answer 21

All the credit goes to you. I didn't have to show you much before you figured it out.

Answer 22

Would you happen to know anything about factoring and simplifying algebraic expressions? I honestly can't understand it for the life of me.

Answer 23

Yup, I know about those

Answer 24

Do you mind helping me understand how to do an expression like this: \[(x+7)^{-1/5}+(x+7)^{-6/5}\]

Answer 25

Well, the first thing you should note is that a negative expression has nothing to do with "negative". In this case, negative exponent means "inverse".

Answer 26

So in other words, you need to write the inverse of \((x+7)^{1/5}\) for example

Answer 27

The inverse of a number is simply 1 over that number. Therefore \(x+7)^{-1/5}\) = \(\large \frac{1}{(x+7)^{1/5}} \)

Answer 28

Okay, so the other would be \[\frac{ 1 }{ (x+7)^{6/5} }\]

Answer 29

Yes

Answer 30

Okay, and then what do I do front there?

Answer 31

*from

Answer 32

Hang on a minute

Answer 33

One thing you can do is this:

Rewrite \(\large \frac{ 1 }{ (x+7)^{6/5} }\) as \(\large\frac{1}{\sqrt[5]{(x+7)^5(x+7)^1}}\)

Answer 34

Let me know if you understand that or not.

Answer 35

the 5 and the 1 equal 6 so it's like 6/5

Answer 36

Okay. Further more, you can split the denominator to this:

\(\large\frac{1}{\sqrt[5]{(x+7)^5} \sqrt[5]{(x+7)^1}}\)

Answer 37

Do you know what \(\sqrt[5]{(x+7)^5}\) simplifies to?

Answer 38

Would it just be \[\sqrt{(x+7)}\]

Answer 39

actually the root cancels as well

Answer 40

leaving just x + 7

Answer 41

Ohhh, so what about the 1/(x+7)1/5?

Answer 42

Basically here's what you're left with now:

\(\large \frac{1}{(x+7)\sqrt[5]{(x+7)}} + \frac{1}{\sqrt[5]{(x+7)}}\)

What do you think you should multiply the second fraction by to create the same denominator?

Answer 43

1/(x+7)?

Answer 44

You can only multiply fractions by equivalents of 1 in this case when you're trying to create the same denominator

Answer 45

In other words, multiply by (x+7)/(x+7)

Answer 46

I hope I didn't lose you.

Answer 47

Sorry, I'm here.

Answer 48

What's the matter?

Answer 49

I went to get something. So, do I multiply the \[\sqrt[5]{(x+7)}\] by (x+7)?

Answer 50

You multiply the entire second fraction by (x+7)/(x+7)

You should be able to interpret what I mean by that.

Answer 51

The demoninators of both fractions are required to be the same before combining them. However, the numerators are not required to be the same.

Answer 52

Okay.

Answer 53

Let me know what you end up with.

Answer 54

Well now it's just a mix of numbers and I'm confused. I have: \[\frac{ 1 }{ (x+7)\sqrt[5]{(x+7)} } +\] the same thing (sorry lost connection

Answer 55

(If I don' reply, I'm eating dinner. But I'll be back! So, you can continue)

Answer 56

You should have ended up with \(\large \frac{ x+8 }{ (x+7)\sqrt[5]{(x+7)} } \) If you had multiplied the second fraction by (x+7)/(x+7), thats what you would have gotten.

Answer 57

You can also simplify the denominator as well

Answer 58

Here's the next step you would perform to do that:

\(\large \frac{ x+8 }{ (x+7)^{5/5}(x+7)^{1/5} }\)

Then afterwards, you would use the rules for multiplying exponents

Answer 59

Eventually, you would end up with \(\large \frac{ x+8 }{ \sqrt[5]{(x+7)^6} }\) as you final simplified form

Answer 60

Thank you so much for your help. I'll be sure to ask you if I have anymore hard questions!

Answer 61

Yes, I'm still here

Answer 62

(: Thank you.

Answer 63

It was such a confusing question!

Answer 64

I actually have another question, if you're up for helping.

Answer 65

You should post it as a new question, then close it immediately that way no one interferes.

Answer 66

Alright!