Question(b) $\frac{\sqrt[5]{128 x^{8}}}{\sqrt[5]{2 x^{2}}}$ $\square$ Question Help: Message instructor

Studdy Solution

STEP 1

What is this asking? Simplify the given expression by combining the fifth roots and simplifying the variables. Watch out! Remember your exponent rules, especially when dividing variables with exponents.
Don't accidentally add the exponents when you should be subtracting!

STEP 2

1. Combine the roots
2. Simplify the numbers
3. Simplify the variables

STEP 3

Alright, let's combine those fifth roots!
Remember, when we divide two radicals with the same index, we can combine them under a single radical.
It's like they're becoming best buddies!
So, we can rewrite our expression as: $\frac{\sqrt[5]{128x^8}}{\sqrt[5]{2x^2}} = \sqrt[5]{\frac{128x^8}{2x^2}}$ This makes things much tidier, doesn't it?

STEP 4

Now, let's simplify the fraction inside the radical.
We have $\frac{128}{2}$ .
That simplifies to 64, because $2 \cdot 64 = 128$ .
So, our expression becomes: $\sqrt[5]{\frac{128x^8}{2x^2}} = \sqrt[5]{64x^8/x^2}$ Isn't that neat?

STEP 5

Now, for the exciting part: simplifying the variables!
We have $x^8$ divided by $x^2$ , which means we subtract the exponents.
Remember, $x^8$ means $x$ multiplied by itself 8 times, and $x^2$ means $x$ multiplied by itself 2 times.
So, when we divide, we're essentially "canceling out" two of those $x$ 's by dividing to one.
This leaves us with $x$ multiplied by itself $8 - 2 = 6$ times, or $x^6$ .
Mathematically, we write: $\frac{x^8}{x^2} = x^{8-2} = x^6$ So, our expression now looks like this: $\sqrt[5]{64x^6}$ Almost there!

STEP 6

We know that $2^6 = 64$ .
So we can rewrite our expression as: $\sqrt[5]{2^6 x^6}$ This is useful because we can now rewrite this as: $\sqrt[5]{(2x)^6}$

STEP 7

We can rewrite our expression using rational exponents: $\sqrt[5]{(2x)^6} = (2x)^{6/5}$ We can rewrite the exponent as a mixed number: $(2x)^{6/5} = (2x)^{1 + 1/5}$ Using the product rule of exponents, we can rewrite this as: $(2x)^{1 + 1/5} = (2x)^1 \cdot (2x)^{1/5}$ And finally, rewriting the fractional exponent back to radical form, we get: $(2x)^1 \cdot (2x)^{1/5} = 2x\sqrt[5]{2x}$

STEP 8

Our final, simplified expression is $2x\sqrt[5]{2x}$ !
We combined the roots, simplified the numbers, and simplified the variables, all while using our awesome exponent rules!

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms