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AlgebraQuestion and Answers: Page 226

Question Number 121127    Answers: 3   Comments: 0

(√(3+2i))=?

$$\sqrt{\mathrm{3}+\mathrm{2}{i}}=? \\ $$

Question Number 121099    Answers: 1   Comments: 0

Let α be a root of x^5 −x^3 +x−2=0 Then prove that [α^6 ]=3 where[λ] denotes greatest integer less than or equal λ

$${Let}\:\alpha\:{be}\:{a}\:{root}\:{of}\:\:{x}^{\mathrm{5}} −{x}^{\mathrm{3}} +{x}−\mathrm{2}=\mathrm{0} \\ $$$${Then}\:{prove}\:{that}\:\:\:\left[\alpha^{\mathrm{6}} \right]=\mathrm{3}\:\:\:\:\:\:\:{where}\left[\lambda\right]\:\:{denotes}\:{greatest}\:{integer} \\ $$$${less}\:{than}\:{or}\:\:{equal}\:\lambda \\ $$

Question Number 121041    Answers: 1   Comments: 0

Σ_(k=1) ^n (−1)^k .k =?

$$\:\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\left(−\mathrm{1}\right)^{\mathrm{k}} .\mathrm{k}\:=?\: \\ $$

Question Number 120993    Answers: 0   Comments: 0

Question Number 120984    Answers: 2   Comments: 0

(1/(1∙4∙7)) + (1/(4∙7∙10))+(1/(7∙10∙13))+...+(1/(25∙28∙31))= ?

$$\frac{\mathrm{1}}{\mathrm{1}\centerdot\mathrm{4}\centerdot\mathrm{7}}\:+\:\frac{\mathrm{1}}{\mathrm{4}\centerdot\mathrm{7}\centerdot\mathrm{10}}+\frac{\mathrm{1}}{\mathrm{7}\centerdot\mathrm{10}\centerdot\mathrm{13}}+...+\frac{\mathrm{1}}{\mathrm{25}\centerdot\mathrm{28}\centerdot\mathrm{31}}=\:? \\ $$

Question Number 120960    Answers: 0   Comments: 0

If x,y,z>0 then prove following inequality (x^2 +2)(y^2 +2)(z^2 +2)≥9(xy+yz+xz)

$${If}\:{x},{y},{z}>\mathrm{0}\:\:{then}\:{prove}\:{following} \\ $$$${inequality} \\ $$$$\left({x}^{\mathrm{2}} +\mathrm{2}\right)\left({y}^{\mathrm{2}} +\mathrm{2}\right)\left({z}^{\mathrm{2}} +\mathrm{2}\right)\geqslant\mathrm{9}\left({xy}+{yz}+{xz}\right) \\ $$

Question Number 120953    Answers: 4   Comments: 0

solve in x∈R ∣ 3x−4 ∣ = x−5

$$\mathrm{solve}\:\mathrm{in}\:\mathrm{x}\in\mathbb{R} \\ $$$$\mid\:\mathrm{3x}−\mathrm{4}\:\mid\:=\:\mathrm{x}−\mathrm{5}\: \\ $$

Question Number 120943    Answers: 3   Comments: 1

∫(x^3 +3)^4 dx=?

$$\int\left({x}^{\mathrm{3}} +\mathrm{3}\right)^{\mathrm{4}} {dx}=? \\ $$

Question Number 120929    Answers: 5   Comments: 2

Question Number 120904    Answers: 1   Comments: 1

Question Number 120882    Answers: 1   Comments: 0

Question Number 120855    Answers: 2   Comments: 0

... elementary calculus... :: α,β are roots of equation of : x^2 −6x−2=0 define :: t_n =α^n −β^n (n≥1) then evaluate : A=((t_(10) −2t_8 )/(2t_9 )) =??? ...m.n.1970...

$$\:\:\:\:\:\:\:\:\:\:\:...\:{elementary}\:\:{calculus}... \\ $$$$\:\:::\:\alpha,\beta\:{are}\:{roots}\:{of}\:\:{equation} \\ $$$$\:\:\:\:\:{of}\::\:{x}^{\mathrm{2}} −\mathrm{6}{x}−\mathrm{2}=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:{define}\:::\:{t}_{{n}} =\alpha^{{n}} −\beta^{{n}} \:\left({n}\geqslant\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:{then}\:\:{evaluate}\:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{A}=\frac{{t}_{\mathrm{10}} −\mathrm{2}{t}_{\mathrm{8}} }{\mathrm{2}{t}_{\mathrm{9}} }\:=??? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{m}.{n}.\mathrm{1970}... \\ $$

Question Number 120839    Answers: 3   Comments: 0

Question Number 120809    Answers: 0   Comments: 0

Find the largest number of positive integers that can be found in such a way that any two of them a and b ( a≠b) satisfy the next inequality ∣a−b∣≥((ab)/(100))

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{number}\:\mathrm{of}\:\mathrm{positive} \\ $$$$\mathrm{integers}\:\mathrm{that}\:\mathrm{can}\:\mathrm{be}\:\mathrm{found}\:\mathrm{in}\:\mathrm{such}\:\mathrm{a}\:\mathrm{way} \\ $$$$\mathrm{that}\:\mathrm{any}\:\mathrm{two}\:\mathrm{of}\:\mathrm{them}\:{a}\:\mathrm{and}\:{b}\:\left(\:{a}\neq{b}\right)\: \\ $$$$\mathrm{satisfy}\:\mathrm{the}\:\mathrm{next}\:\mathrm{inequality}\:\mid{a}−{b}\mid\geqslant\frac{{ab}}{\mathrm{100}} \\ $$

Question Number 120780    Answers: 2   Comments: 2

solve x^2^x =−(1/2)

$${solve}\:{x}^{\mathrm{2}^{{x}} } =−\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 120711    Answers: 3   Comments: 0

show that 1+2+3+4...=((−1)/8)

$${show}\:{that}\:\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}...=\frac{−\mathrm{1}}{\mathrm{8}} \\ $$

Question Number 120636    Answers: 0   Comments: 23

selective Binomial

$${selective}\:{Binomial} \\ $$

Question Number 120601    Answers: 1   Comments: 0

Find the range of values of x for which the expansion of the binomial (2 − 3x)^(−4) is valid. I need help with explanation please

$$\:\mathrm{Find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{for}\: \\ $$$$\:\mathrm{which}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\mathrm{the}\:\mathrm{binomial} \\ $$$$\:\left(\mathrm{2}\:−\:\mathrm{3}{x}\right)^{−\mathrm{4}} \:\mathrm{is}\:\mathrm{valid}.\: \\ $$$$\:{I}\:{need}\:{help}\:{with}\:{explanation}\:{please} \\ $$

Question Number 120583    Answers: 0   Comments: 0

Question Number 120558    Answers: 3   Comments: 0

Question Number 120556    Answers: 0   Comments: 2

Question Number 120545    Answers: 1   Comments: 2

((√x))^(x/( (√x))) = (1/( (√2))) x=?

$$\left(\sqrt{\boldsymbol{{x}}}\right)^{\frac{\boldsymbol{{x}}}{\:\sqrt{\boldsymbol{{x}}}}} \:=\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\:\:\:\:\boldsymbol{{x}}=? \\ $$

Question Number 120529    Answers: 1   Comments: 1

Question Number 120480    Answers: 4   Comments: 0

show that ∀ n ∈N^∗ Σ_(k=1) ^n k(n−k)=(((n−1)(n+1))/6)

$${show}\:{that}\:\forall\:{n}\:\in\mathbb{N}^{\ast} \\ $$$$\sum_{{k}=\mathrm{1}} ^{{n}} {k}\left({n}−{k}\right)=\frac{\left({n}−\mathrm{1}\right)\left({n}+\mathrm{1}\right)}{\mathrm{6}} \\ $$

Question Number 120475    Answers: 1   Comments: 0

Question Number 120472    Answers: 0   Comments: 0

A=5x23^(−) ^6 show that A≡x−4[7] deduct the value of x for which A is divisible by 7

$${A}=\overline {\mathrm{5}{x}\mathrm{23}}\:^{\mathrm{6}} \:{show}\:{that}\:{A}\equiv{x}−\mathrm{4}\left[\mathrm{7}\right] \\ $$$${deduct}\:{the}\:{value}\:{of}\:{x}\:{for}\:{which}\:{A}\:{is}\: \\ $$$${divisible}\:{by}\:\mathrm{7} \\ $$

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