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AlgebraQuestion and Answers: Page 228

Question Number 115908    Answers: 1   Comments: 2

what is the cofficient of x^2 (1+x)(1+2x)(1+4x)......(1+2^n x)

$${what}\:{is}\:{the}\:{cofficient}\:{of}\:{x}^{\mathrm{2}} \\ $$$$\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+\mathrm{2}{x}\right)\left(\mathrm{1}+\mathrm{4}{x}\right)......\left(\mathrm{1}+\mathrm{2}^{{n}} {x}\right) \\ $$

Question Number 115906    Answers: 1   Comments: 0

find all pairs of integers (x,y) such that x^3 +y^3 =(x+y)^2

$${find}\:{all}\:{pairs}\:{of}\:{integers}\:\left({x},{y}\right)\:{such}\:{that} \\ $$$${x}^{\mathrm{3}} +{y}^{\mathrm{3}} =\left({x}+{y}\right)^{\mathrm{2}} \\ $$

Question Number 115897    Answers: 2   Comments: 1

Given a_n = (√(1 + (1 − (1/n))^2 )) + (√(1 + (1 + (1/n))^2 )) The value of Σ_(n=1) ^(2015) ((4/a_n )) is ...

$$\mathrm{Given} \\ $$$${a}_{{n}} \:=\:\sqrt{\mathrm{1}\:+\:\left(\mathrm{1}\:−\:\frac{\mathrm{1}}{{n}}\right)^{\mathrm{2}} }\:+\:\sqrt{\mathrm{1}\:+\:\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{{n}}\right)^{\mathrm{2}} } \\ $$$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\underset{{n}=\mathrm{1}} {\overset{\mathrm{2015}} {\sum}}\left(\frac{\mathrm{4}}{{a}_{{n}} }\right)\:\mathrm{is}\:... \\ $$

Question Number 115858    Answers: 2   Comments: 0

What are all ordered pairs of real number (x,y) for which 5^(y−x) (x+y) = 1 and (x+y)^(x−y) = 5

$${What}\:{are}\:{all}\:{ordered}\:{pairs}\:{of}\:{real} \\ $$$${number}\:\left({x},{y}\right)\:{for}\:{which}\: \\ $$$$\mathrm{5}^{{y}−{x}} \:\left({x}+{y}\right)\:=\:\mathrm{1}\:{and}\:\left({x}+{y}\right)^{{x}−{y}} \:=\:\mathrm{5} \\ $$

Question Number 115857    Answers: 1   Comments: 0

What are all real values of p for which the inequality −3<((x^2 +px−2)/(x^2 −x+1))<2 is satisfied by all real values of x

$${What}\:{are}\:{all}\:{real}\:{values}\:{of}\:{p}\:{for} \\ $$$${which}\:{the}\:{inequality}\: \\ $$$$−\mathrm{3}<\frac{{x}^{\mathrm{2}} +{px}−\mathrm{2}}{{x}^{\mathrm{2}} −{x}+\mathrm{1}}<\mathrm{2}\:{is}\:{satisfied}\: \\ $$$${by}\:{all}\:{real}\:{values}\:{of}\:{x} \\ $$

Question Number 115871    Answers: 1   Comments: 1

Question Number 115815    Answers: 1   Comments: 0

Montrer que ∀(a,b,c)∈(R_+ ^∗ )^3 (1/(a^2 +bc))+(1/(b^2 +ac))+(1/(c^2 +ab))≤(1/2)((1/(ab))+(1/(bc))+(1/(ac)))

$$\mathrm{Montrer}\:\mathrm{que}\:\:\forall\left(\mathrm{a},\mathrm{b},\mathrm{c}\right)\in\left(\mathbb{R}_{+} ^{\ast} \right)^{\mathrm{3}} \\ $$$$\frac{\mathrm{1}}{\mathrm{a}^{\mathrm{2}} +\mathrm{bc}}+\frac{\mathrm{1}}{\mathrm{b}^{\mathrm{2}} +\mathrm{ac}}+\frac{\mathrm{1}}{\mathrm{c}^{\mathrm{2}} +\mathrm{ab}}\leqslant\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{ab}}+\frac{\mathrm{1}}{\mathrm{bc}}+\frac{\mathrm{1}}{\mathrm{ac}}\right) \\ $$

Question Number 115773    Answers: 0   Comments: 0

((1−x)/( (√(x^2 −2x+5)))) = (3/5)

$$\:\frac{\mathrm{1}−{x}}{\:\sqrt{{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{5}}}\:=\:\frac{\mathrm{3}}{\mathrm{5}} \\ $$

Question Number 115564    Answers: 1   Comments: 0

Question Number 115544    Answers: 1   Comments: 1

Given x^2 +12(√x) = 5 then x+2(√x) ?

$${Given}\:{x}^{\mathrm{2}} +\mathrm{12}\sqrt{{x}}\:=\:\mathrm{5} \\ $$$${then}\:{x}+\mathrm{2}\sqrt{{x}}\:? \\ $$

Question Number 115484    Answers: 1   Comments: 3

find the value of ((111)/(1+1+1)) , ((222)/(2+2+2)) , ((333)/(3+3+3)), ((444)/(4+4+4)) ((555)/(5+5+5)), ((666)/(6+6+6)) , ((777)/(7+7+7)) , ((888)/(8+8+8)) ((999)/(9+9+9))

$${find}\:{the}\:{value}\:{of}\: \\ $$$$\frac{\mathrm{111}}{\mathrm{1}+\mathrm{1}+\mathrm{1}}\:,\:\frac{\mathrm{222}}{\mathrm{2}+\mathrm{2}+\mathrm{2}}\:,\:\frac{\mathrm{333}}{\mathrm{3}+\mathrm{3}+\mathrm{3}},\:\frac{\mathrm{444}}{\mathrm{4}+\mathrm{4}+\mathrm{4}} \\ $$$$\frac{\mathrm{555}}{\mathrm{5}+\mathrm{5}+\mathrm{5}},\:\frac{\mathrm{666}}{\mathrm{6}+\mathrm{6}+\mathrm{6}}\:,\:\frac{\mathrm{777}}{\mathrm{7}+\mathrm{7}+\mathrm{7}}\:,\:\frac{\mathrm{888}}{\mathrm{8}+\mathrm{8}+\mathrm{8}} \\ $$$$\frac{\mathrm{999}}{\mathrm{9}+\mathrm{9}+\mathrm{9}} \\ $$

Question Number 115436    Answers: 1   Comments: 0

(√((a)^(1/3) +(b)^(1/3) ))=(1/( (√(b−a×s^3 ))))(((−s^2 ×(a^2 )^(1/3) )/2)+s((ab))^(1/3) +(b^2 )^(1/3) ) what is s?

$$\sqrt{\sqrt[{\mathrm{3}}]{{a}}+\sqrt[{\mathrm{3}}]{{b}}}=\frac{\mathrm{1}}{\:\sqrt{{b}−{a}×{s}^{\mathrm{3}} }}\left(\frac{−{s}^{\mathrm{2}} ×\sqrt[{\mathrm{3}}]{{a}^{\mathrm{2}} }}{\mathrm{2}}+{s}\sqrt[{\mathrm{3}}]{{ab}}+\sqrt[{\mathrm{3}}]{{b}^{\mathrm{2}} }\right) \\ $$$${what}\:{is}\:{s}? \\ $$

Question Number 115333    Answers: 1   Comments: 3

Question Number 115325    Answers: 0   Comments: 5

Question Number 115218    Answers: 0   Comments: 0

Show that ∀n∈N, ∀u_0 ,u_1 ,...,u_n ,v_0 ,v_1 ,...v_n ∈C ∀k≤n; u_k =Σ_(i=0) ^k ((k),(i) )v_i ⇔∀k≤n; v_k =Σ_(i=0) ^k (−1)^(k−1) ((k),(i) )u_i

$$\mathrm{Show}\:\mathrm{that}\:\forall\mathrm{n}\in\mathbb{N},\:\forall\mathrm{u}_{\mathrm{0}} ,\mathrm{u}_{\mathrm{1}} ,...,\mathrm{u}_{\mathrm{n}} ,\mathrm{v}_{\mathrm{0}} ,\mathrm{v}_{\mathrm{1}} ,...\mathrm{v}_{\mathrm{n}} \in\mathbb{C} \\ $$$$\forall\mathrm{k}\leqslant\mathrm{n};\:\mathrm{u}_{\mathrm{k}} =\sum_{\mathrm{i}=\mathrm{0}} ^{\mathrm{k}} \begin{pmatrix}{\mathrm{k}}\\{\mathrm{i}}\end{pmatrix}\mathrm{v}_{\mathrm{i}} \Leftrightarrow\forall\mathrm{k}\leqslant\mathrm{n};\:\mathrm{v}_{\mathrm{k}} =\sum_{\mathrm{i}=\mathrm{0}} ^{\mathrm{k}} \left(−\mathrm{1}\right)^{\mathrm{k}−\mathrm{1}} \begin{pmatrix}{\mathrm{k}}\\{\mathrm{i}}\end{pmatrix}\mathrm{u}_{\mathrm{i}} \\ $$

Question Number 115136    Answers: 1   Comments: 0

Question Number 115135    Answers: 1   Comments: 1

Question Number 115023    Answers: 2   Comments: 0

solve { ((7x−5y+3z=6)),((2x+4y−5z=−5)),((9x−8y+2z=−1)) :}. Find x+y+z

$${solve}\:\begin{cases}{\mathrm{7}{x}−\mathrm{5}{y}+\mathrm{3}{z}=\mathrm{6}}\\{\mathrm{2}{x}+\mathrm{4}{y}−\mathrm{5}{z}=−\mathrm{5}}\\{\mathrm{9}{x}−\mathrm{8}{y}+\mathrm{2}{z}=−\mathrm{1}}\end{cases}.\:{Find}\:{x}+{y}+{z}\: \\ $$

Question Number 115018    Answers: 2   Comments: 0

If 9^x +9^(−x) = 3^(2+x) +3^(2−x) −20, then 27^x +27^(−x) =?

$${If}\:\mathrm{9}^{{x}} +\mathrm{9}^{−{x}} \:=\:\mathrm{3}^{\mathrm{2}+{x}} +\mathrm{3}^{\mathrm{2}−{x}} \:−\mathrm{20},\:{then}\: \\ $$$$\mathrm{27}^{{x}} +\mathrm{27}^{−{x}} \:=? \\ $$

Question Number 114956    Answers: 0   Comments: 5

Question Number 114919    Answers: 0   Comments: 0

Question Number 114912    Answers: 0   Comments: 0

Question Number 114917    Answers: 0   Comments: 3

Question Number 114809    Answers: 1   Comments: 1

{ ((1−((12)/(y+3x))=(2/( (√x))))),((1+((12)/(y+3x))=(6/( (√y))))) :}

$$\begin{cases}{\mathrm{1}−\frac{\mathrm{12}}{{y}+\mathrm{3}{x}}=\frac{\mathrm{2}}{\:\sqrt{{x}}}}\\{\mathrm{1}+\frac{\mathrm{12}}{{y}+\mathrm{3}{x}}=\frac{\mathrm{6}}{\:\sqrt{{y}}}}\end{cases} \\ $$

Question Number 114806    Answers: 1   Comments: 0

Question Number 114799    Answers: 1   Comments: 0

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