Answer:
12. Option A is correct
13. Option A is correct
14. Option C is correct
15. Option D is correct
16. Option A is correct
Step-by-step explanation:
12) Lowest Common Denominator of
[tex]\frac{p+3}{p^2+7p+10} \,\, and \,\, \frac{p+5}{p^2+5p+6}[/tex]
We should find the factors of denominators and then find the LCM of the denominators.Finding LCD is same as finding LCM.
Factors of p^2+7p+10 = p^2 +2p +5p+10 = p(p+2)+5(p+2) = (p+5)(p+2)
Factors of p^2+5p+6 = p^2+2p+3p+6 = p(p+2)+3(p+2) = (p+3) (p+2)
Now, rewriting the above equation with factors and finding the LCM
[tex]\frac{p+3}{(p+5)(p+2)} \,\, and \,\, \frac{p+5}{(p+3)(p+2)}\\[/tex]
LCM of (p+5)(p+2) and (p+3)(p+2) = (p+5)(p+3)(p+2)
The LCD is (p+5)(p+3)(p+2).
So, Option A is correct.
13. Divide
[tex]\frac{40x}{64y} \,\,by\,\, \frac{5x}{8y}[/tex]
by stands for division. The equation can be written as:
[tex]\frac{40x}{64y}\div\frac{5x}{8y}[/tex]
Division sign changed into multiplication, we take reciprocal of second term i.e,
[tex]\frac{40x}{64y}*\frac{8y}{5x}\\\\Solving\\\frac{40x*8y}{64y*5x}\\\\\frac{320xy}{320xy} \\\\1[/tex]
So, Option A is correct.
14. Simplify:
[tex]\frac{x+2}{x^2-6x-16} \div \frac{1}{9x} \\[/tex]
Factors of x^2-6x-16= x^2 -8x +2x -16 = x(x-8)+2(x-8) = (x-8)(x+2)
Putting factors in the above equation and changing division sign with multiplication we get,
[tex]\frac{x+2}{(x-8)(x+2)} * \frac{9x}{1}\\\frac{9x}{x-8}[/tex]
So, Option C is correct.
15. Simplify
[tex]\frac{4}{\frac{1}{4}-\frac{5}{2}}[/tex]
Solving denominator,
Taking LCM of 4 and 2 and subtracting we get
[tex]\frac{4}{\frac{1-(5*2)}{4}}\\\frac{4}{\frac{1-10}{4}}\\\frac{4}{\frac{-9}{4}}\\\frac{4*4}{-9}\\\frac{16}{-9} \,\,or\,\,\\\frac{-16}{9}[/tex]
Option D is correct.
16. Simplify:
[tex]\frac{7x+42}{x^2+13x+42}[/tex]
Making factors of x^2+13x+42= x^2 +6x+7x+42 = x(x+6)+7(x+6) = (x+7)(x+6)
Taking 7 common from numerator and putting factors in denominator we get,
[tex]\frac{7(x+6)}{(x+7)(x+6)}\\\\Cancelling\,\, x+6 \,\, from\,\, numerator \,\, and\,\, \\\\\frac{7}{(x+7)}[/tex]
Option A is correct.
