Question

2.Simplify the given expression to rational exponent form, justify each step by identifying the properties of rational exponents used. All work must be shown.

Answer 1

\[\frac{1}{\sqrt[3]{x^{-6}}}\]

Answer 2

@AravindG can you help explain how he got it

Answer 3

I thought you wanted to understand how to simplify the expression.

Answer 4

Its same as question?

Answer 5

@deezgurls, do you want an explanation of how to simplify the expression?

Answer 6

Yeah, I don't know how to simplify it D: Sorry for the hassle

Answer 7

So first, understand the rule

\[x^{-a} = \frac{1}{x^a}\]

\(x^{-a}\) can be read as the "inverse of x to the a".

Answer 8

In other words, the negative exponent means "inverse".

Answer 9

Okay

Answer 10

So the first property so apply is "inverse property of exponents", which after application we get:

\[\dfrac{1}{\sqrt[3]{\dfrac{1}{x^6}}}\]

Answer 11

From here, there are a few different rules we can use. First of all understand that

\[\sqrt[3]{1^3} = 1\]
\[\sqrt[6]{1^6} = 1\]

Answer 12

Actually let's do this, rewrite the fraction as:

\[\dfrac{1}{\dfrac{\sqrt[3]{1}}{\sqrt[3]{x^6}}}\]

We can do that by way of the following rule:
\[\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}\]

Answer 13

okay, I wrote it down

Answer 14

Next, we simplify the numerator of the inner fraction according to the rule above to get:

\[\dfrac{1}{\dfrac{1}{\sqrt[3]{x^6}}}\]

Answer 15

okay

Answer 16

There's another rule we can utilize to attempt to simplify further:
\[x^{ab} = (x^{a})^b\]

In this case:
\[x^6 = x^{(2)(3)} = (x^2)^3\]

Answer 17

Alright

Answer 18

So we can re-write the fraction as:
\[\dfrac{1}{\dfrac{1}{\sqrt[3]{(x^2)^3}}}\]

Answer 19

And we see yet another rule we can use. In general:

\[\large\sqrt[n]{x^n} = x\]

Answer 20

The root and exponent cancel if they have the same base.

Answer 21

This is because

\[\large\sqrt[n]{x^n} = x^{\frac{n}{n}} = x^1 = x\]

Answer 22

So now we have just:
\[\dfrac{1}{\dfrac{1}{x^2}}\]

Which can be re-written as :

\[1 \div \frac{1}{x^2}\]

We can do this by way of the definition of division:

\[\frac{a}{b} = a \div b\]

Answer 23

Finally, we have the reciprocal rule, which utliizes the fact that division is merely the inverse of multiplication. In other words:

\[a \div b = a \times \frac{1}{b}\]

So we have:

\[1 \div \frac{1}{x^2} = 1 \times x^2\]

Answer 24

In other words, the final result is just

\[x^2\]

Answer 25

so the answer is x2?

Answer 26

okay now i'm confused

Answer 27

Basically, negative exponent doesn't mean the answer will be negative.

Answer 28

Remember, negative exponent means "inverse"

Answer 29

Also, remember, that we had to reduce the expression. And we used the rules of exponents and algebra to do so.

Answer 30

Alright, thank you so much!