Could I get some help with my math homework?

Question

I cannot figure out how to simplify the following equation:

(3x+1)^(4/5) - (3x+1)^2
--------------------------------
(3x+1)^(7/5)


Can anyone give me some help?

This being summer homework, I honestly don't know how far to take the simplification, but I guess I'll just go as far as possible...

Answer 1

Hi,

The answer depends on how far your teacher expects you to go to simplify it.

3x+1)^(4/5) - (3x+1)^2
------------------------------ can also be split into 2 fractions as:
(3x+1)^(7/5)

(3x+1)^(4/5) (3x+1)^2
---------------..-..-------------...
(3x+1)^(7/5)...(3x+1)^(7/5)

Simplifying each fraction separately by subtracting their exponents gives:

........1
---------------..-.(3x+1)^(3/5) <== this is simplified.
(3x+1)^(3/5)

However, if someone wanted to write this all as one fraction, you'd have to make a common denominator of (3x+1)^(3/5) for both parts.

........1
---------------..-.(3x+1)^(3/5)
(3x+1)^(3/5)

........1...............(3x+1)^(6/5)
---------------..-..-----------------
(3x+1)^(3/5)....(3x+1)^(3/5)

These combine into:

.1..-..(3x+1)^(6/5)
----------------------- <== This could be considered simplified.
(3x+1)^(3/5)

If you didn't want a radical (fractional exponent) in the denominator. you'd have to multiply top and bottom both by (3x+1)^(2/5). This will rationalize the denominator.

[1.-.(3x+1)^(6/5)](3x+1)^(2/5)
-----------------------------------
(3x+1)^(3/5)*(3x+1)^(2/5)

(3x+1)^(2/5).-.(3x+1)^(8/5)]
----------------------------------- <== this could also be considered
.............3x+1.................................simplified


I hope that helps!! :-)

Answer 2

(3x+1)^(4/5) - (3x+1)^2
---------------------------------- =
(3x+1)^(7/5)

= [(3x + 1)^(4/5) - (3x + 1)^2] (3x + 1)^(-7/5) =
= (3x + 1)^(4/5) * (3x + 1)^(-7/5) - (3x + 1)^2 *
* (3x + 1)^(-7/5) = (3x + 1)^(4/5 - 7/5) +
+ (3x + 1)^(2 - 7/5) =
= (3x + 1)^(-3/5) + (3x + 1)^(10/5 - 7/5) =
= (3x + 1)^(-3/5) + (3x + 1)^(3/5) =

1
-------------------- + (3x + 1)^(3/5) =
(3x + 1)^(3/5)

... 1 + (3x + 1)^(6/5)
= ---------------------------
... (3x + 1)^(3/5)

Answer 3

[(3x + 1)^(4/5) - (3x + 1)^(10/5)]/(3x + 1)^7/5

(3x + 1)^(4/5)[1 - (3x + 1)^(6/5)]/(3x + 1)^(7/5)

[1 - (3x + 1)^(6/5)]/(3x + 1)^(3/5)