27^-4/3
How do I solve this?

- anonymous

27^-4/3
How do I solve this?

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- Nnesha

\[\huge\rm x^{-m} = \frac{ 1 }{ x^m }\]
if there is a negative exponent then you should flip the fraction

- anonymous

@Nnesha okay- what do I do after that?

- anonymous

\[27^{-\frac{ 4 }{ 3 }} = \frac{ 1 }{ 27^\frac{ 4 }{ 3 } } = \frac{ 1 }{ \left( 27^\frac{ 1 }{ 3 } \right)^4 }\]

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## More answers

- anonymous

And\[x^\frac{ 1 }{ 3 } = \sqrt[3]{x}\]

- anonymous

I'm a bit confused with that second equation. How/when would I use that?

- anonymous

Well, any time you have a fractional exponent, the denominator of the fraction is the root that is required. In other words,\[x^\frac{ n }{ m } = \sqrt[m]{x^n}\]And you have a fractional exponent in your question.

- anonymous

Put simply, you can simplify \(27^\frac{1}{3}\)

- anonymous

So does -4/3 get plugged into that or does 1/3

- anonymous

I already converted my equation

- anonymous

I'm sorry. I don't understand. Are you able to simplify\[\frac{ 1 }{ \left( 27^\frac{ 1 }{ 3 } \right)^4 }\]

- anonymous

I don't think so, but I'm asking if I use the exponent -4/3 or the exponent 1/3 to plug into that rational expression

- mathstudent55

This problem involves two exponent rules.
1. a negative exponent
2. a fractional exponent
Here are the rules you need:
Negative exponent:
\(\Large a ^{-n} = \dfrac{1}{a^n} \)
Fractional exponent:
\(\Large a^{\frac{m}{n}} = \sqrt[n] {a^m} = \left( \sqrt[n] a \right)^m \)

- triciaal

|dw:1439436483720:dw|

- triciaal

|dw:1439436579962:dw|

- mathstudent55

Let's use the negative exponent rule first.
|dw:1439436586340:dw|

- triciaal

|dw:1439436666466:dw|

- anonymous

We break the fractional exponent into two parts and use the "power to a power" rule\[27^\frac{ 4 }{ 3 } = 27^{\frac{ 1 }{ 3 }\times4} = \left( 27^\frac{ 1 }{ 3 } \right)^4\]

- anonymous

I understand that so far but which exponent do I use? The already converted exponent?

- mathstudent55

The negative exponent is already taken care of.
Now let's take care of the fractional exponent:
|dw:1439436644180:dw|

- anonymous

Order of operations. Calculate what 27^(1/3) is and raise that number to the 4th power.

- mathstudent55

The fractional exponent has been taken care of, and now you have a cubic root and an exponent.
Take the cubic root of 27.
Then raise it to the 4th power.

- triciaal

@ospreytriple the order will not matter you will get the same result the only operation is really multiplication

- anonymous

I'm sorry...I'm on the app so the links or equations y'all are trying to show me aren't working. I can see the computerized version...

- anonymous

9^4?

- anonymous

@triciaal , I hope you'll agree it's a lot easier to take the cube of 27 and raise that result to the 4th power than it is to raise 27 to the 4th power and then take the cube root. Technically, you are correct of course, but there are practical concerns to consider.

- anonymous

Or 3^4

- anonymous

Yes. 3^4. The cube root of 27 is 3.

- mathstudent55

|dw:1439437072181:dw|

- triciaal

of course I agree that is fastest for this problem but I was addressing what you said about the order of operations

- anonymous

I was following the order of operations as I wrote it (with brackets).

- anonymous

So 81?

- anonymous

@achara.the.blasian, you're almost there. What is 3^4?

- anonymous

Well, don't forget, it's 1/81.

- anonymous

Ohhhhh okay. So what do I do with that?

- mathstudent55

|dw:1439437425872:dw|

- anonymous

That's your answer. The question requires you to simplify the given expression. And you have. Good job.

- anonymous

Thank you so much everyone

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