Simplify š„+4
š„ā4
ā
5š„+20
š„
2 ā16Simplify š„+4
š„ā4
ā
5š„+20
š„
2 ā16
To simplify the expression š„+4 / š„ā4 - (5š„+20 / š„^2 -16), we need to find a common denominator.
The first fraction already has a common denominator of (š„ā4), so we don't need to do anything with that term.
For the second fraction, we can factor the denominator as (š„+4)(š„-4). Therefore, the common denominator for both fractions is (š„+4)(š„ā4).
Now, let's rewrite the fractions with the common denominator:
š„+4 / š„ā4 = (š„+4)(š„+4) / (š„ā4)(š„+4)
5š„+20 / š„^2 -16 = 5(š„+4) / (š„ā4)(š„+4)
Now, we can combine the fractions:
(š„+4)(š„+4) / (š„ā4)(š„+4) - 5(š„+4) / (š„ā4)(š„+4)
To subtract the fractions, we need to have a common denominator, which we already have. So, we can combine the numerators:
(š„+4)(š„+4) - 5(š„+4) / (š„ā4)(š„+4)
Next, we can simplify the numerator:
(š„+4)(š„+4) - 5(š„+4) = š„^2 + 8š„ + 16 - 5š„ - 20
Now, let's combine like terms in the numerator:
š„^2 + 8š„ + 16 - 5š„ - 20 = š„^2 + 3š„ - 4
Now, our simplified expression is:
(š„^2 + 3š„ - 4) / (š„ā4)(š„+4)
To simplify the expression š„+4/š„ā4ā5š„+20/š„^2ā16, we can follow these steps:
Step 1: Factor the denominators š„^2ā16 and š„ā4
The denominator š„^2ā16 is a difference of squares and can be factored as (š„+4)(š„ā4).
The denominator š„ā4 is already factored.
Step 2: Find the least common denominator (LCD) of the fractions.
The LCD of the fractions is (š„+4)(š„ā4).
Step 3: Rewrite each fraction with the LCD.
The first fraction š„+4/š„ā4 can be rewritten as (š„+4)(š„+4)/(š„+4)(š„ā4).
The second fraction ā5š„+20/š„^2ā16 can be rewritten as -5(š„ā4)/(š„+4)(š„ā4).
Step 4: Simplify the numerators.
The first fraction's numerator remains (š„+4)(š„+4).
The second fraction's numerator simplifies to -5(š„ā4) = -5š„ + 20.
Step 5: Combine the fractions.
Now, we can combine the two fractions by keeping the LCD as the denominator:
(š„+4)(š„+4)/(š„+4)(š„ā4) - 5š„ + 20/(š„+4)(š„ā4).
Step 6: Simplify the numerator of the first fraction.
The numerator of the first fraction simplifies to (š„+4)(š„+4) = š„^2+8š„+16.
Step 7: Combine the fractions and simplify.
Now, we can combine the fractions by adding the numerators:
(š„^2+8š„+16)/(š„+4)(š„ā4) - 5š„ + 20/(š„+4)(š„ā4).
Finally, we simplify:
(š„^2+8š„+16 - 5š„ + 20)/(š„+4)(š„ā4).
This simplifies to:
(š„^2+3š„+36)/(š„+4)(š„ā4).