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2 answers:

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I have to rewrite the expressions because they look very confusing.

1) The given fractions to add are:

4p² + 6 2p - 2
------------ + ------------
 4p² 10p²

2) The statements to asses are:

A) The least common denominator is 20p
.
B) The first fraction rewritten with the LCD is (20p² +30) / 20p²

C) The second fraction rewritten with the LCD is (4p - 4) / 20p

D) The sum is (24p² -4p +30 ) / 20p² = (12p-2p+15) / 10p^2

E) The sum is a rational expression.

This is the result for each statement.


A) The least common denominator is 20p

FALSE: THE LCD IS 20p², because it the least expression that contains all the prime factors of both denominators: 4p² and 10p.

See that you cannot divide 20p by 4p² but you can divide 20p² by 4p² and 10 p. So, 20p is not a common denominator.
.
B) The first fraction rewritten with the LCD is (20p² +30) / 20p²

TRUE: you get this expression from the first fraction by multiplying both numerator and denominator by 20p² / 4p² = 5:

(4p² + 6)×5 = 20p² + 30

(4p²)×5 = 20p²

=>

20p² + 30
--------------
 20p²

C) The second fraction rewritten with the LCD is (4p - 4) / 20p

FALSE: you have to multiply both numerator an denominator by 20p² / 10p = 2p

(2p - 2) × 2p = 4p² - 4p

10p × 2p = 20p²

So, the second fraction rewritten with the LCD is:

4p² - 4p
-----------
 20p²

D) The sum is (24p² - 4p +30 ) / 20p² = (12p-2p+15) / 10p²

TRUE


 20p² + 30 4p² - 4p 24p² - 4p + 30
-------------- + --------------- = ----------------------
 20p² 20p² 20p²

And when you simplify the last fraction dividing by 2 you get:

12p² - 2p + 15
-----------------------
 10p²

E) The sum is a rational expression.

TRUE:

The sum of rational number has the closure property which means that by adding rational expression you will always ge a rational expression.