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AlgebraQuestion and Answers: Page 300

Question Number 67371 Answers: 1 Comments: 0

Question Number 67333 Answers: 0 Comments: 0

evaluate Σ_(n=0) ^(+∞) (1/((1+8n)^2 ))

$${evaluate}\:\underset{{n}=\mathrm{0}} {\overset{+\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{8}{n}\right)^{\mathrm{2}} } \\ $$

Question Number 67337 Answers: 1 Comments: 0

m(p+1)=a ((q(p+1))/(r+1))=b & ((p(p+1))/(q+1))=c and ((r(p+1))/(m+1))=d find either of p,q,r,m in terms of a,b,c,d.

$${m}\left({p}+\mathrm{1}\right)={a} \\ $$$$\frac{{q}\left({p}+\mathrm{1}\right)}{{r}+\mathrm{1}}={b}\:\:\&\:\:\frac{{p}\left({p}+\mathrm{1}\right)}{{q}+\mathrm{1}}={c} \\ $$$${and}\:\:\frac{{r}\left({p}+\mathrm{1}\right)}{{m}+\mathrm{1}}={d} \\ $$$${find}\:{either}\:{of}\:{p},{q},{r},{m}\:{in}\:{terms}\:{of} \\ $$$${a},{b},{c},{d}. \\ $$

Question Number 67244 Answers: 0 Comments: 7

Which of the series converge and which diverge? Check by the limit comparison test. 1) Σ_(n=2) ^∞ ((1+n ln(n))/(n^2 +5)) 2) Σ_(n=1) ^∞ ((ln(n))/n^(3/2) ) 3) Σ_(n=3) ^∞ (1/(ln(lnn))) 4) Σ_(n=1) ^∞ (1/(n (n)^(1/n) )) ??

$${Which}\:{of}\:{the}\:{series}\:{converge}\:{and}\: \\ $$$${which}\:{diverge}?\:{Check}\:{by}\:{the}\:{limit} \\ $$$${comparison}\:{test}. \\ $$$$\left.\mathrm{1}\right)\:\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}+{n}\:{ln}\left({n}\right)}{{n}^{\mathrm{2}} +\mathrm{5}} \\ $$$$\left.\mathrm{2}\right)\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{ln}\left({n}\right)}{{n}^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$$$\left.\mathrm{3}\right)\:\underset{{n}=\mathrm{3}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{ln}\left({lnn}\right)} \\ $$$$\left.\mathrm{4}\right)\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{n}\:\left({n}\right)^{\frac{\mathrm{1}}{{n}}} }\:\: \\ $$$$?? \\ $$

Question Number 67208 Answers: 1 Comments: 6

Find the times in a day when the hour′s, minute′s and second′s hand of a clock occupy the same angular position. [old question reposted]

$${Find}\:{the}\:{times}\:{in}\:{a}\:{day}\:{when} \\ $$$${the}\:{hour}'{s},\:{minute}'{s}\:{and}\:{second}'{s} \\ $$$${hand}\:{of}\:{a}\:{clock}\:{occupy}\:{the}\:{same} \\ $$$${angular}\:{position}. \\ $$$$\left[{old}\:{question}\:{reposted}\right] \\ $$

Question Number 67167 Answers: 4 Comments: 2

solve for real x and y:[a,b∈R] a. { ((x^3 +1=y^3 )),((x^2 +1=y^2 )) :} b. { ((x^3 +x^2 +1=y^3 )),((x^2 +x+1=y^2 )) :} c. { ((x^3 +y^2 =9xy)),((x^2 +y^3 =8xy)) :} d. { ((ax+by=2ab)),((x^2 +y^2 =4abxy)) :}

Question Number 67136 Answers: 2 Comments: 0

factorize 2x^3 −1

$${factorize}\:\mathrm{2}{x}^{\mathrm{3}} −\mathrm{1} \\ $$

Question Number 67025 Answers: 0 Comments: 3

Question Number 66995 Answers: 0 Comments: 4

Question Number 66969 Answers: 1 Comments: 1

Question Number 66863 Answers: 0 Comments: 0

x^3 (b−x)^2 +aex(x−b)+e^2 =0 solve for x.

$$\:\:\:{x}^{\mathrm{3}} \left({b}−{x}\right)^{\mathrm{2}} +{aex}\left({x}−{b}\right)+{e}^{\mathrm{2}} =\mathrm{0}\: \\ $$$${solve}\:{for}\:{x}. \\ $$

Question Number 66857 Answers: 0 Comments: 3

Question Number 66855 Answers: 2 Comments: 0

simplify ((p^2 +2pq+q^2 )/(p^3 −pq^2 +p^2 q−q^3 ))

$${simplify} \\ $$$$\frac{{p}^{\mathrm{2}} +\mathrm{2}{pq}+{q}^{\mathrm{2}} }{{p}^{\mathrm{3}} −{pq}^{\mathrm{2}} +{p}^{\mathrm{2}} {q}−{q}^{\mathrm{3}} } \\ $$

Question Number 66833 Answers: 0 Comments: 0

Somewhere , there are scorpio, snake and mouse. We ascertain that : Every morning , each snake eats a mouse . Every afternoon ,each scorpio kills a snake. And every night , each mouse eats a scorpio. Two weeks passed and we find that there was remaining only one animal . 1) If that animal is a snake , how many snake were there two week ago? 2)If that animal is a scorpio , how many scorpio were there two weeks ago? 3)If that animal is a mouse ,how many mouse were there two weeks ago? 4)If there are remaining one animal of each breed , How many were they for each breed.

$${Somewhere}\:,\:{there}\:{are}\:{scorpio},\:{snake}\:{and}\:{mouse}. \\ $$$${We}\:{ascertain}\:{that}\:: \\ $$$${Every}\:{morning}\:,\:{each}\:{snake}\:{eats}\:{a}\:{mouse}\:. \\ $$$${Every}\:{afternoon}\:,{each}\:{scorpio}\:\:{kills}\:{a}\:{snake}. \\ $$$${And}\:{every}\:{night}\:,\:{each}\:{mouse}\:{eats}\:{a}\:{scorpio}. \\ $$$${Two}\:{weeks}\:{passed}\:{and}\:{we}\:{find}\:{that}\:{there}\:{was}\:{remaining}\:{only}\:{one}\:{animal}\:. \\ $$$$\left.\mathrm{1}\right)\:{If}\:{that}\:{animal}\:{is}\:{a}\:{snake}\:,\:{how}\:{many}\:{snake}\:{were}\:{there}\:{two}\:{week}\:{ago}? \\ $$$$\left.\mathrm{2}\right){If}\:{that}\:{animal}\:{is}\:{a}\:{scorpio}\:,\:{how}\:{many}\:{scorpio}\:{were}\:{there}\:{two}\:{weeks}\:{ago}? \\ $$$$\left.\mathrm{3}\right){If}\:{that}\:{animal}\:{is}\:{a}\:{mouse}\:,{how}\:{many}\:{mouse}\:{were}\:{there}\:\:{two}\:{weeks}\:{ago}? \\ $$$$\left.\mathrm{4}\right){If}\:{there}\:{are}\:{remaining}\:{one}\:{animal}\:{of}\:{each}\:{breed}\:,\:{How}\:{many}\:{were}\:{they}\:{for}\:{each}\:{breed}. \\ $$

Question Number 66827 Answers: 0 Comments: 3

Question Number 66769 Answers: 1 Comments: 2

simplify ((x+4)/(x−4))−((5x+20)/(x^2 −16))

$${simplify} \\ $$$$\frac{{x}+\mathrm{4}}{{x}−\mathrm{4}}−\frac{\mathrm{5}{x}+\mathrm{20}}{{x}^{\mathrm{2}} −\mathrm{16}} \\ $$

Question Number 66760 Answers: 2 Comments: 2

By writing your answer in the form a^y simplify (3^(5x) ×5^(2x) ×3^(−x) ÷5^(−2x) )^(1/4)

$${By}\:{writing}\:{your}\:{answer}\:{in}\:{the} \\ $$$${form}\:{a}^{{y}} \:{simplify} \\ $$$$\left(\mathrm{3}^{\mathrm{5}{x}} ×\mathrm{5}^{\mathrm{2}{x}} ×\mathrm{3}^{−{x}} \boldsymbol{\div}\mathrm{5}^{−\mathrm{2}{x}} \right)^{\frac{\mathrm{1}}{\mathrm{4}}} \\ $$

Question Number 66743 Answers: 0 Comments: 4

Each month a store owner can spend at most $100,000 on PC′s and laptops. A PC costs the store owner $1000 and a laptop costs him $1500. Each PC is sold for a profit of $400 while a laptop is sold for a profit of $700. The store owner estimates that at least 15 PC′s but no more than 80 are sold each month. He also estimates that the number of laptops sold is at most half the PC′s. How many PC′s and how many laptops should be sold in order to maximize the profit?

$$\mathrm{Each}\:\mathrm{month}\:\mathrm{a}\:\mathrm{store}\:\mathrm{owner}\:\mathrm{can}\:\mathrm{spend}\:\mathrm{at} \\ $$$$\mathrm{most}\:\$\mathrm{100},\mathrm{000}\:\mathrm{on}\:\mathrm{PC}'\mathrm{s}\:\mathrm{and}\:\mathrm{laptops}.\:\mathrm{A} \\ $$$$\mathrm{PC}\:\mathrm{costs}\:\mathrm{the}\:\mathrm{store}\:\mathrm{owner}\:\$\mathrm{1000}\:\mathrm{and}\:\mathrm{a} \\ $$$$\mathrm{laptop}\:\mathrm{costs}\:\mathrm{him}\:\$\mathrm{1500}.\:\mathrm{Each}\:\mathrm{PC}\:\mathrm{is}\:\mathrm{sold} \\ $$$$\mathrm{for}\:\mathrm{a}\:\mathrm{profit}\:\mathrm{of}\:\$\mathrm{400}\:\mathrm{while}\:\mathrm{a}\:\mathrm{laptop}\:\mathrm{is}\:\mathrm{sold} \\ $$$$\mathrm{for}\:\mathrm{a}\:\mathrm{profit}\:\mathrm{of}\:\$\mathrm{700}.\:\mathrm{The}\:\mathrm{store}\:\mathrm{owner}\:\mathrm{estimates} \\ $$$$\mathrm{that}\:\mathrm{at}\:\mathrm{least}\:\mathrm{15}\:\mathrm{PC}'\mathrm{s}\:\mathrm{but}\:\mathrm{no}\:\mathrm{more}\:\mathrm{than} \\ $$$$\mathrm{80}\:\mathrm{are}\:\mathrm{sold}\:\mathrm{each}\:\mathrm{month}.\:\mathrm{He}\:\mathrm{also}\:\mathrm{estimates} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{laptops}\:\mathrm{sold}\:\mathrm{is}\:\mathrm{at}\:\mathrm{most} \\ $$$$\mathrm{half}\:\mathrm{the}\:\mathrm{PC}'\mathrm{s}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{PC}'\mathrm{s}\:\mathrm{and}\:\mathrm{how} \\ $$$$\mathrm{many}\:\mathrm{laptops}\:\mathrm{should}\:\mathrm{be}\:\mathrm{sold}\:\mathrm{in}\:\mathrm{order}\:\mathrm{to} \\ $$$$\mathrm{maximize}\:\mathrm{the}\:\mathrm{profit}? \\ $$

Question Number 66715 Answers: 0 Comments: 4

if f(x)=((ln (x+(√(1+x^2 ))))/(√(1+x^2 ))) f^(−1) (x)=?

$${if}\:{f}\left({x}\right)=\frac{\mathrm{ln}\:\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)}{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }} \\ $$$${f}^{−\mathrm{1}} \left({x}\right)=? \\ $$

Question Number 66619 Answers: 1 Comments: 0

solve for x,y∈R ((√(1+x^2 ))/(ln (x+(√(1+x^2 )))))=((√(1+y^2 ))/(ln (y+(√(1+y^2 )))))

$${solve}\:{for}\:{x},{y}\in{R} \\ $$$$\frac{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}{\mathrm{ln}\:\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)}=\frac{\sqrt{\mathrm{1}+{y}^{\mathrm{2}} }}{\mathrm{ln}\:\left({y}+\sqrt{\mathrm{1}+{y}^{\mathrm{2}} }\right)} \\ $$

Question Number 66546 Answers: 0 Comments: 6

Question Number 66508 Answers: 0 Comments: 3

Question Number 66382 Answers: 0 Comments: 14

Give me any Quintic, i shall solve it. For sure! At^5 +Bt^4 +Ct^3 +Dt^2 +Et+F=0 wont even assume A=1, or B=0. but if A+C+E=B+D+F then my formula dont work but then obviously t=−1 is a root!

$${Give}\:{me}\:{any}\:{Quintic},\:{i}\:{shall}\:{solve} \\ $$$${it}.\:{For}\:{sure}! \\ $$$${At}^{\mathrm{5}} +{Bt}^{\mathrm{4}} +{Ct}^{\mathrm{3}} +{Dt}^{\mathrm{2}} +{Et}+{F}=\mathrm{0} \\ $$$${wont}\:{even}\:{assume}\:{A}=\mathrm{1},\:{or}\:{B}=\mathrm{0}. \\ $$$${but}\:{if}\:{A}+{C}+{E}={B}+{D}+{F}\: \\ $$$${then}\:{my}\:{formula}\:{dont}\:{work} \\ $$$${but}\:{then}\:{obviously}\:{t}=−\mathrm{1}\:{is}\:{a}\:{root}! \\ $$

Question Number 66381 Answers: 0 Comments: 0

Question Number 66302 Answers: 0 Comments: 3

solved the general quintic, despite whatever proof that it cant be solved in a simple way!

$${solved}\:{the}\:{general}\:{quintic}, \\ $$$${despite}\:{whatever}\:{proof}\:{that}\:{it} \\ $$$${cant}\:{be}\:{solved}\:{in}\:{a}\:{simple}\:{way}! \\ $$

Question Number 66290 Answers: 1 Comments: 0

(x^2 )^(1/(√3)) + x^(√3) − 392 = 0

$$\:\sqrt[{\sqrt{\mathrm{3}}}]{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\:+\:\boldsymbol{\mathrm{x}}^{\sqrt{\mathrm{3}}} \:−\:\mathrm{392}\:=\:\mathrm{0} \\ $$

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