Question

Simplify the expression and rationalize the demonimator when appropriate, please show steps : >

Answer 1

-3v²t

Explanation:

`\sqrt[3]{3t^4v^2}  \sqrt[3]{-9t^-1v^4}`

=
`\sqrt[3]{3t} \sqrt[3]{v^2t} \sqrt[3]{-9v^4t^-1}`

=
`\sqrt[3]{3t} (-\sqrt[3]{9v^4/t)} \sqrt[3]{v^2t}`

= -
`\sqrt[3]{3t}  \sqrt[3]{v^2 9v^4/t}t`

= -
`\sqrt[3]{3}  \sqrt[3]{9t}  \sqrt[3]{v^6}`

= -
`\sqrt[3]{3*9}v^2t`

= -
`\sqrt[3]{27} v^2t`

= -3v²t

Answer 2

`=-3tv^2`

Explanation:

So we have the expression:

`\sqrt[3]{3t^4v^2}\sqrt[3]{-9t^(-1)v^4}`

To simplify, combine the two radicals. Since they have the same index, we can combine them. Thus:

`=\sqrt[3]{(3t^4v^2)(-9t^(-1)v^4)}`

Combine like terms:

`=\sqrt[3]{(3\cdot -9)(t^4\cdot t^(-1))(v^2\cdot v^4)}`

Multiply. When multiplying the exponents, simply add the exponents:

`=\sqrt[3]{-27t^3v^6}`

Now, simplify. Note that -27 can be written as (-3)^3. t^3 can be written as (t)^3 and v^6 can be written as (v^2)^3. Thus:

`=\sqrt[3]{(-3)^3(t)^3(z^2)^3}`

Combine them all under one exponent:

`=\sqrt[3]{(-3tv^2)^3}`

Cancel out the cube root:

`=-3tv^2`

And this is the simplest it can get.

And we are done :)

Edit: Typo