Expert: Sherman D. - 5/16/2004

Question
so I have more questions then usuall, give steps and answer.

1. simplify (f^8/g^3)^8

2. simplify (d^3e^2)^3

3. simplify (w^3x)^6

4. use synthetic division to evaluate
  f(c)=2c^3-3c^2-2c-14 when c=2

5. factor completley 3x^3-3x^2+6x

6. solve x^3+5x^2=0

7. divide (x^3-3x-3)divided by(x+2)

8. divide using synthetic division
 (2x^4-2x^3-4x-5)divided by(x-2)

9. find all zeros of the function
  f(x)=-5(-4x+2)(-4x-8)(-9x-6)

10. evaluate 8^-1/3^ (^^ means -1/3 is tiny)

11. simplify 27^4/3^

12. the number of bacteria present in a culture after t minutes is given as B=100c^kt^ if there are 2139 bacteria after 9 minutes find k

13. find an equation for the inverse of the relation y=5x+3

14. evaluate log^3 729

15. solve for x to the nearest hundreath
   6.13^x=40

16. simplify the rational expression          n^2+2n-24/n^2-11n+28

17. divide (x^2+9x+20/x^2-25)divided by(x+4/x-4)

18. simplify (-3x+3/-3x)+(2x-3/-3x)

19. add (2/x+8)+(3/x-8)

20. simplify 1-4/3/1+3/5

21. simplify -3/x-1/3/x+4 first - is for whole top fraction last 4 is added to entire second fraction, its not part of bottum fraction

22.solve (x^2/x+3)=(9/x+3)

23.solve (p/p+1)+(p/p+4)=1

24. subtract (-2x^3+6x)-(-4x-5+5x^3)

25. factor 3x^7-6x^5-24x^3

26.evaluate without using a calculator
  log^7 1/49

27. evaluate log^6 30 to three decimal places

28. write the equation log^9 243=5/2 in exponential form

29.solve the equation 35log^6 x=18

30. solve the equation 6^-0.2x^-3=7

31.multiply (7y^2/5).(15x/6y)

32. find the vertical and horizonal asymptotes of
   f(x)=4-2x/x-3)

33. write mthe standard form of the equation of the circle with radius 9 and center at (0,0)

thanks very very much!!!

Answer
I did all of this in about 2 hours and 20 minutes. The only reason why it took me that long, because i was having trouble with the synthetic problems, because i made some mistakes that i had to discover and correct, also because i had to try and figure out what you meant to type.

raising powers = multipling them
dividing powers = subtracting them
multipling powers = adding them
for log problems, i used http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/calctopic1/logs.html

1.) simplify (f^8/g^3)^8
(f^8^8)/(g^3^8)
becomes
(f^(8 * 8))/(g^(3 * 8))
becomes
(f^64)/(g^24)

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2.) simplify (d^3e^2)^3
(d^3^3)*(e^2^3)
becomes
(d^(3 * 3))/(e^(2 * 3))
(d^9)/(e^6)

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3.) simplify (w^3x)^6
(w^3^6) * x^6
(w^(3 * 6)) * x^6
w^18x^6

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4.) use synthetic division to evaluate
f(c) = 2c^3 - 3c^2 - 2c - 14 when c = 2
This will be tough for it to show up right.

2'|'2'-3'-2'|'-14
''|''''4''2'|'''0
--------------------
''|'2''1''0'|'-14

So you can say if you were to use c = 2 as a factor, you would get

2c^2 + c R -14
or you can say
2c^2 + c - (14/(c - 2))

Anyway, if you were actually 2 in for c, you get

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5.) factor completley 3x^3-3x^2+6x
3x^3 - 3x^2 + 6x
3x(x^2 - x + 2)

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6.) solve x^3+5x^2=0
x^3 + 5x^2 = 0
x^2(x + 5) = 0
x = 0 and -5

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7.) divide (x^3-3x-3)divided by(x+2)
(x^3 - 3x - 3)/(x + 2)
Using synthetic division

-2'|'1''0'-3'|'-3
'''|'''-2''4'|'-2
--------------------
'''|'1'-2''1'|'-5

x^2 - 2x + 1 - (5/(x + 2))

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8.) divide using synthetic division
(2x^4 - 2x^3 - 4x - 5) divided by (x - 2)

2'|'2'-2'0'-4'|'-5
''|''''4'4''8'|''8
---------------------
''|'2''2'4''4'|''3

so the answer is
2x^3 + 2x^2 + 4x + 4 R 3
or you can say
2x^3 + 2x^2 + 4x + 4 + (3/(x - 2))
or you can say
2x^3 + 2x^2 - (5/(x - 2))

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9.) find all zeros of the function
f(x) = -5(-4x + 2)(-4x - 8)(-9x - 6)
becomes
f(x) = -5(-2(2x - 1)) * (-4(x + 2)) * (-3(3x + 2))
f(x) = (-5 * -2 * -4 * -3) * ((2x - 1)(x + 2)(3x + 2))
f(x) = (10 * 12) * (2x - 1)(x + 2)(3x + 2)
f(x) = 120(2x - 1)(x + 2)(3x + 2)
All zeros are
x = (1/2), -2 and (-2/3)


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10.) evaluate 8^-1/3^ (^^ means -1/3 is tiny)
It would be simplier to type and for me to understand if you type it like this
8^(-1/3)
Trust me, i will understand if if you do it that way
anyway
8^(-1/3)
becomes
1/(8^(1/3))
1/(cbrt(8))
ANS : (1/2)

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11.) simplify 27^4/3
27^(4/3)
becomes
cbrt(27^4)
cbrt(531441)
however if you meant
(27^4)/3, then
(531441)/3
177147

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12.) the number of bacteria present in a culture after t minutes is given as B = 100c^kt if there are 2139 bacteria after 9 minutes find k.

Does "c" have a constant value, or did you mean "e", because "e" has a constant value.

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13.) find an equation for the inverse of the relation
y = 5x + 3
switch the y and x and solve for y
so
y = 5x + 3
x = 5y + 3
subtract 3 from both sides
x - 3 = 5y
divide both sides by 5
y = (x - 3)/5
or
y = (1/5)x - (3/5)


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14.) evaluate log^3 729
(log 729)/(log 3) = 6

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15. solve for x to the nearest hundreath
  6.13^x=40
If by this you do mean
6.13^x = 40, then
log(6.13)40 = x
becomes
(log 40)/(log 6.13) = 2.03
Thats if you mean 6.13^x and not 6 * 13^x

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16.) simplify the rational expression       
(n^2 + 2n - 24)/(n^2 - 11n + 28)
BETTER
Factor all
((n + 6)(n - 4))/((n - 7)(n - 4))

(n - 4)s cancel out, leaving you with

ANS: (n + 6)/(n - 7)

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17.) divide (x^2+9x+20/x^2-25)divided by(x+4/x-4)
((x^2 + 9x + 20)/(x^2 - 25)) / ((x + 4)/(x - 4))
Multiply top half by reciprocal of bottom half
((x^2 + 9x + 20)/(x^2 - 25)) * ((x - 4)/(x + 4))
Multiply across
((x^2 + 9x + 20)(x - 4)) / ((x^2 - 25)(x + 4))
Factor all
((x + 5)(x + 4)(x - 4))/((x - 5)(x + 5)(x + 4))
since there is a (x + 5)(x + 4) on top and bottom, they cancel out leaving you with

ANS: (x - 4)/(x - 5)

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18.) simplify (-3x+3/-3x)+(2x-3/-3x)
((-3x + 3)/(-3x)) + ((2x - 3)/(-3x))
same denominator, so this gives you
((-3x + 3) + (2x - 3))/(-3x)
(-3x + 3 + 2x - 3)/(-3x)
(-x)/(-3x)
((-1)/(-3)) * (x/x)
ANS: (1/3)

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19.) add (2/x+8)+(3/x-8)
(2/(x + 8)) + (3/(x - 8))
Multiply everything by (x + 8)(x - 8)
(2(x - 8) + 3(x + 8))/((x + 8)(x - 8))
(2x - 16 + 3x + 24)/(x^2 - 8x + 8x - 64)
(5x + 8)/(x^2 - 64)

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20.) simplify 1-4/3/1+3/5
I am assuming i have this right and if not, don't blame me because you can't separate your values with ().
1 - 4 / 3 / 1 + 3 / 5
Using PEMDAS
1 - ((4/3)/1) + (3/5)
1 - (4/3) + (3/5)
Multiply everything by 15
(15 - 20 + 9)/15
(-5 + 9)/15
(4/15)

However if you meant
(1 - (4/3)) / (1 + (3/5))
then Multiply top by 3 and bottom by 5 and you get
((3 - 4)/3)/((5 + 3)/5)
(-1/3)/(8/5)
Multiply top half by reciprocal of bottom half
(-1/3)*(5/8)
(-5/24)

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21.) simplify -3/x-1/3/x+4 first - is for whole top fraction last 4 is added to entire second fraction, its not part of bottum fraction

If by this you mean

(-3/(x - 1)) / (3/(x + 4))
multiply top half by reciprocal of bottom half and you get
(-3/(x - 1)) * ((x + 4)/3)
Multiply straight across
(-3(x + 4))/(3(x - 1))
becomes
(-3/3) * (x + 4)/(x - 1)
becomes
-1 * (x + 4)/(x - 1)
becomes
(-(x + 4))/(x - 1))
becomes
ANS:(-x - 4)/(x - 1)

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22.) solve (x^2/(x + 3)) = (9/(x + 3))
cross multiply
9(x + 3) = x^2(x + 3)
divide both sides by (x + 3)
x^2 = 9
square root both sides
x = 3 or -3
not certain if -3 can be used since -3 would give you an undefined value, but since it would make both sides undefined and therefore equal, that is why i am not certain.

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23.) solve (p/(p + 1)) + (p/(p + 4)) = 1
Multiply everything by (p + 1)(p + 4)
p(p + 4) + p(p + 1) = (p + 1)(p + 4)
p^2 + 4p + p^2 + p = p^2 + 4p + p + 4
(1 + 1)p^2 + (4 + 1)p = p^2 + 5p + 4
2p^2 + 5p = p^2 + 5p + 4
subtract p^2, 5p, and 4 from both sides
(2 - 1)p^2 + (5 - 5)p - 4 = 0
p^2 - 4 = 0
(p - 2)(p + 2) = 0
p = 2 or -2

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24.) subtract (-2x^3 + 6x) - (-4x - 5 + 5x^3)
(-2x^3 + 6x) - (-4x - 5 + 5x^3)
-2x^3 + 6x + 4x + 5 - 5x^3
(-2 - 5)x^3 + (6 + 4)x + 5
(-2 + (-5))x^3 + 10x + 5
-7x^3 + 10x + 5

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25.) factor 3x^7 - 6x^5 - 24x^3

3x^7 - 6x^5 - 24x^3
3x^3(x^4 - 2x^2 - 8)
3x^3(x^2 - 4)(x^2 + 2)
3x^3(x - 2)(x + 2)(x^2 + 2)
now if you have reached the point of complex values then
3x^3(x - 2)(x + 2)(x + isqrt(2))(x - isqrt(2))
whereas "i" is the sqrt(-1)

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26.)evaluate without using a calculator
log^7 1/49
log(base 7)(1/49)
becomes
(log(1/49))/(log 7) = -2

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27.) evaluate log^6 30 to three decimal places
log(base 6)30
becomes
(log 30)/(log 6) = 1.898

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28.) write the equation log^9 243=5/2 in exponential form
If by this you mean
log(base 9)243 = (5/2)
9^(5/2) = 243
There you go.

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29.) solve the equation 35log^6 x = 18
If by this you mean 35log(base 6)x = 18, then i can help
35log(base 6)x = 18 becomes
log(base 6)(x^35)
becomes
6^18 = x^35
x^35 = 6^18
35thrt both sides and i get
ANS : about 2.513

I went to quickmath and they gave me many other solutions.
To get it to work, you would have to type it like this
(log(x^35))/(log(6)) = 18 exactly like that or it will not work.

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30.) solve the equation 6^-0.2x^-3=7
you will have to tell me what you meant by separating the values. For ex: 6^-0, the zero can't be negative. If you meant -x instead of 0 and if by .2x^-3 you meant to multiply, then i can help.

If you meant
6^(-.2) * x^(-3) = 7
becomes
6^(-1/5) * (1/(x^3)) = 7
becomes
(1/(6^(1/5))) * (1/(x^3)) = 7
multiply straight across
(1/(6^(1/5) * x))) = 7
multiply both sides by (6^(1/5))x
1 = 7 * (6^(1/5))x
divide both sides by 7
(1/7) = 6^(1/5)x
divide both sides by 6^(1/5)
(1/(7 * 6^(1/5))) = x
x = about .099832 or .1
you can also say
x = (7 * 6^(1/5))^-1
which gives you about
x = 10.01678^-1
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31.)multiply (7y^2/5).(15x/6y)
((7y^2)/5) * ((15x)/(6y))
Multiply across
(105xy^2)/(30y)
becomes
(105/30) * x * (y^(2 - 1))
(7/2) * x * y^1
ANS : (7xy)/2

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32.) find the vertical and horizonal asymptotes of
  f(x)=4-2x/x-3)
If by this you meant
f(x) = (4 - 2x)/(x - 3)
becomes
f(x) = (-2x + 4)/(x - 3)
the vertical asymptote is what x cannot be, which is 3, as for the horizontal asymptote. To find that, switch the x and y and solve for y, and what x cannot be is your horizontal asymptote, so

f(x) = (-2x + 4)/(x - 3)
same as
y = (-2x + 4)/(x - 3)
becomes
x = (-2y + 4)/(y - 3)
multiply both sides by y - 3
xy - 3x = -2y + 4
add 3x and 2y to both sides
xy + 2y = 3x + 4
factor left side
y(x + 2) = 3x + 4
divide both sides by (x + 2)
y = (3x + 4)/(x + 2)

so the answers are

VA = 3
HA = -2

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33.) write the standard form of the equation of the circle with radius 9 and center at (0,0)
Formula is
x^2 + y^2 = r^2
Ans: x^2 + y^2 = 81

PLEASE IN THE FUTURE, IF YOU WISH FOR ME TO SOLVE YOUR PROBLEMS MORE ACTURATLY, YOU NEED TO WORK ON YOUR TYPING AND SEPARATING YOUR VALUES USING (). I CAN ALSO ANSWER YOUR PROBLEMS QUICKER IF I CAN UNDERSTAND WHAT YOU ARE TYPING. ALSO TRY NOT TO MAKE ANY MISTAKES.