Question

describe how to transform (^3 sqrt x^4)^5 into an expression with a rational exponent. make sure you respond with complete sentences

Answer 1

Still need help? Note: it might be better if you try to write your expression with the Equation writer.

Answer 2

\[\left( \sqrt[3]{X ^{4}} \right)^{5}\]

Answer 3

this is the equation please help

Answer 4

Which part confuses you?

Answer 5

i wouldn't know how to transform it into an expression with a rational exponent.

Answer 6

first, do you know how to write

\[\sqrt[3]{x} or \sqrt{x} \]
as an exponent?

Answer 7

First, if you have something like:

\[y = \sqrt[3]{x}\]
that means that:
\[y^{3} = x\]
Are you with me so far?

Answer 8

yes so far i was kind of behind with my class so anything helps please

Answer 9

Also, do you know how to simplify an expression like this:

\[(x^{3})^{4}\]

Answer 10

nope please explain

Answer 11

what about:
\[x^{3}*x^{3}\]
Do you know what this equals?

Answer 12

yes \[X^{3}\]

Answer 13

?

Answer 14

Not quite...

Answer 15

Here's an example:
What's 8*8?

Answer 16

64

Answer 17

Because
\[8 = 2^{3}\]
And 8*8 is the same as:
\[2^{3}*2^{3}\]

Answer 18

So I'll give you a second shot:

\[x^3*x^3 = ?\]

Answer 19

\[X^{6}\]

Answer 20

Bingo!

Answer 21

took me a min little slow

Answer 22

Another way of saying that is that:

\[x^a*x^b = x^{a+b}\]

Answer 23

oh okay

Answer 24

And, with exponents,
\[(x^a)^3 = x^a*x^a*x^a = x^{a+a+a} = x^{3a}\]

Answer 25

So,
\[x^a*x^b = x^{a+b}\]
BUT,
\[(x^a)^b = x^{ab}\]

Answer 26

Are you getting a hang of this?

Answer 27

now that your taking your time to explain it I'm starting to understand it

Answer 28

Ok, great!

Answer 29

Now, the hardest part about that first expression you gave me was the "cubed root". You have any idea how to deal with those??

Answer 30

nope

Answer 31

First, I want you to simplify these two expressions while I type something about cubed roots: \[x^3*2x^2\] and
\[(x^4)^7\]

Answer 32

\[2x^{5}\]

Answer 33

First, lets say I give you something like:

\[(x^a)^3\]
a.) first, simplify this
b.) if I say that:
\[(x^a)^3= x\]
what do you think "a" equals.

Answer 34

\[\left(x ^{47} \right)\]

Answer 35

a equals 4

Answer 36

Whoops, not quite. You got the first question right (I even threw in that extra "2" to trip you up).

Answer 37

Look back at that second question -- (x^4)^7

Answer 38

\[\left( X^{27} \right)\]

Answer 39

You're almost there, except remember: the exponent equals 4*7,

Answer 40

i mean 28 sorry

Answer 41

Ok, great!

Answer 42

Here's another one:
If
\[x^a = x\]
Then what is "a" equal to?

Answer 43

\[x^{2}\]

Answer 44

So you're saying:
\[x^2 = x\]
?

Answer 45

i believe so.

Answer 46

Now, that IS true when x = 0 or when x =1.... but what about for ALL x's

Answer 47

Well, try plugging in a number for "x" to see. For instance, try out "x=3"

Answer 48

will it be 9?

Answer 49

You're right: 3^2 = 9. Now, let's go back to that original question:
\[x^a = x\]
If "a" was equal to "2" then that would mean that:
\[x^2 = x\]
And, if we plugged in "3" for "x" then you would have:
\[3^2 = 3\]
But you just said that 3^2 = 9 (which is true). Therefore,
\[3^2 \neq 3\]

Answer 50

This might sound boring but I swear this part is important for when you turn the "cubed root" into an exponent...

Answer 51

sorry was in restroom

Answer 52

Next time you can just say "I was busy".
:)

Answer 53

sorry but thank you

Answer 54

I'll give you a second to catch up -- I have to take some medicine myself (I'm sick).

Answer 55

okay thank you and sorry to hear that i just got over a cold myself.

Answer 56

Ok, now back to that question I gave you earlier:

\[x^a = x\]

Answer 57

\[x^{a2}\]

Answer 58

Not quite. But the good news is that it's easier than it looks.

Answer 59

a = 1

Because
\[x^1 \]
is just another way of writing "x"

Answer 60

ohh yea duh i should of got that right

Answer 61

Because if:
\[x^3 = x*x*x\]
\[x^2 = x*x\]
Then
\[x^1 = x\]

Answer 62

Now, back to roots, what is:
\[\sqrt[3]{x}*\sqrt[3]{x}*\sqrt[3]{x}\]

Answer 63

\[\sqrt[27]{x}\]

Answer 64

\[\sqrt[27]{X}^{3}\]

Answer 65

How about this one:

\[\sqrt{x}*\sqrt{x}\]

Answer 66

Hint: It's easier than it looks...

Answer 67

And it always helps if you plug in some numbers:

\[\sqrt{16}*\sqrt{16} = ?\]

Answer 68

but how do you always know which numbers to plug in?

Answer 69

No, whenever you see an "x" that means that it could stand for ANY number. Thus, I can choose any number I want just to test my answers. There's nothing special about the "16" I chose--I chose it because the square root of 16 is easy to calculate (it's just "4"). Similarly, when talking about cube roots, I would probably chose an a number like "27" because I know from the top of my head that the cube root of 27 equals 3.

Answer 70

Now, if I gave you something like:

\[9 = \sqrt{x}\]
You can't just plug in any number you want for "x" --you have to SOLVE for x.
But, if you just have regular expression (i.e. with no "equals" sign) then you can always plug in a number for "x" and see if your solution makes sense.

Answer 71

Remember how I earlier wrote that:
\[y = \sqrt[3]{x}\]
means that:
\[y ^{3} = x\]?
Well, that means that:
\[\sqrt[3]{y^3} = x\]

Answer 72

yes

Answer 73

Actually, sometimes it's most helpful to work backwards. Because it will all make sense after I give you the answer...
First, to rewrite a cubed root as an exponent:

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

Answer 74

and, because:
\[(x^a)^b = x^{a*b}\]
That means that you can take any type of root such as:
\[\sqrt[7]{x^5}\]
and rewrite it as:
\[(x^5)^\frac{ 1 }{ 7 }\]
which than can be simplified to:
\[x^\frac{ 5 }{ 7 }\]

Answer 75

So to go back to your original question:

\[(\sqrt[3]{x^4})^5 = ((x^4)^\frac{ 1 }{ 3 })^5 = (x^\frac{ 4 }{ 3 })^5 = x^\frac{ 20 }{ 3 }\]
Thus your answer is "x raised to the twenty thirds power" or similarly, "x to the 6 and two thirds power"

Answer 76

i truly thank you so much you are the biggest help I've ever had truly better than my own teachers but then again I'm in online so i guess I'm my own teacher and i suck lol

Answer 77

Well, exponents can be confusing... Good luck with the rest of your work!

Answer 78

thanks