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with a,b∈R prove that ((a+(√3)bi))^(1/3) +((a−(√3)bi))^(1/3) has always real value and find this value (or a way how to find). examples: a=15, b=((28)/9) ⇒ result=5 a=6, b=((35)/9) ⇒ result=4 a=−24, b=((80)/9) ⇒ result=4

$${with}\:{a},{b}\in{R}\:{prove}\:{that} \\ $$$$\sqrt[{\mathrm{3}}]{{a}+\sqrt{\mathrm{3}}{bi}}+\sqrt[{\mathrm{3}}]{{a}−\sqrt{\mathrm{3}}{bi}} \\ $$$${has}\:{always}\:{real}\:{value}\:{and}\:{find}\:{this} \\ $$$${value}\:\left({or}\:{a}\:{way}\:{how}\:{to}\:{find}\right). \\ $$$${examples}: \\ $$$${a}=\mathrm{15},\:{b}=\frac{\mathrm{28}}{\mathrm{9}}\:\:\Rightarrow\:{result}=\mathrm{5} \\ $$$${a}=\mathrm{6},\:{b}=\frac{\mathrm{35}}{\mathrm{9}}\:\:\Rightarrow\:{result}=\mathrm{4} \\ $$$${a}=−\mathrm{24},\:{b}=\frac{\mathrm{80}}{\mathrm{9}}\:\:\Rightarrow\:{result}=\mathrm{4} \\ $$

Question Number 78853 Answers: 4 Comments: 0

∫_0 ^(π/2) ∫_0 ^(π/2) ((sin(x)+sin(y))/(x+y))dxdy?

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{sin}\left(\mathrm{x}\right)+\mathrm{sin}\left(\mathrm{y}\right)}{\mathrm{x}+\mathrm{y}}\mathrm{dxdy}? \\ $$

Question Number 78851 Answers: 0 Comments: 1

Question Number 78829 Answers: 0 Comments: 8

given f(x)=f(x+4) ∀x∈R and ∫_5 ^7 f(x)dx=p . what is ∫_2 ^(10) f(x)dx?

$$\mathrm{given}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}+\mathrm{4}\right)\:\forall\mathrm{x}\in\mathbb{R} \\ $$$$\mathrm{and}\:\underset{\mathrm{5}} {\overset{\mathrm{7}} {\int}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\mathrm{p}\:.\:\mathrm{what}\:\mathrm{is}\: \\ $$$$\underset{\mathrm{2}} {\overset{\mathrm{10}} {\int}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}? \\ $$$$ \\ $$

Question Number 78828 Answers: 0 Comments: 3

lim_(x→∞) (2+sin x).lnx = ?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{2}+\mathrm{sin}\:{x}\right).{lnx}\:=\:? \\ $$

Question Number 78827 Answers: 0 Comments: 2

what is inverse function of y= ((5x^5 −3x^3 +x)/(4x^4 −2x^2 +1))

$$\mathrm{what}\:\mathrm{is}\:\mathrm{inverse}\:\:\mathrm{function} \\ $$$$\mathrm{of}\:\mathrm{y}=\:\frac{\mathrm{5x}^{\mathrm{5}} −\mathrm{3x}^{\mathrm{3}} +\mathrm{x}}{\mathrm{4x}^{\mathrm{4}} −\mathrm{2x}^{\mathrm{2}} +\mathrm{1}} \\ $$

Question Number 78815 Answers: 1 Comments: 0

show that sin((2π)/5)=sin((3π)/5)

$$\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{sin}\frac{\mathrm{2}\pi}{\mathrm{5}}=\mathrm{sin}\frac{\mathrm{3}\pi}{\mathrm{5}} \\ $$

Question Number 78814 Answers: 1 Comments: 1

Question Number 78820 Answers: 1 Comments: 0

please what is the fomula to determinate the equations of bissectors in triangle?

$$\mathrm{please}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{fomula}\:\mathrm{to}\: \\ $$$$\mathrm{determinate}\:\mathrm{the}\:\mathrm{equations}\:\mathrm{of}\: \\ $$$$\mathrm{bissectors}\:\mathrm{in}\:\mathrm{triangle}? \\ $$

Question Number 78800 Answers: 1 Comments: 3

Simplify: _3 (√(3+((10)/3)(√(1/3))i))+ _3 (√(3−((10)/3)(√(1/3))i))

$${Simplify}: \\ $$$$\underset{\mathrm{3}} {\:}\sqrt{\mathrm{3}+\frac{\mathrm{10}}{\mathrm{3}}\sqrt{\frac{\mathrm{1}}{\mathrm{3}}}{i}}+\underset{\mathrm{3}} {\:}\sqrt{\mathrm{3}−\frac{\mathrm{10}}{\mathrm{3}}\sqrt{\frac{\mathrm{1}}{\mathrm{3}}}{i}} \\ $$

Question Number 78799 Answers: 1 Comments: 3

Question Number 78797 Answers: 1 Comments: 0

Show that: ∫_( 0) ^( ∞) (x^3 /(e^x − 1)) dx = (π^4 /(15))

$$\mathrm{Show}\:\mathrm{that}:\:\:\:\:\int_{\:\mathrm{0}} ^{\:\infty} \:\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{e}^{\mathrm{x}} \:−\:\mathrm{1}}\:\mathrm{dx}\:\:\:\:=\:\:\:\frac{\pi^{\mathrm{4}} }{\mathrm{15}} \\ $$

Question Number 78794 Answers: 1 Comments: 0

f(x + (1/x)) = ((x^6 + 1)/(27)) f(x) = ...

$${f}\left({x}\:+\:\frac{\mathrm{1}}{{x}}\right)\:\:=\:\:\frac{{x}^{\mathrm{6}} \:+\:\mathrm{1}}{\mathrm{27}} \\ $$$${f}\left({x}\right)\:\:=\:\:... \\ $$

Question Number 78791 Answers: 1 Comments: 0

lim_(x→∞) (√(2+3x−x^2 )) −(√(x^2 −2x+2)) ?

$$ \\ $$$$ \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\sqrt{\mathrm{2}+\mathrm{3x}−\mathrm{x}^{\mathrm{2}} }\:−\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{2x}+\mathrm{2}}\:? \\ $$

Question Number 78785 Answers: 1 Comments: 0

(1+sin (π/7))^(3−cos 2x) = (sin (π/(14))+cos (π/(14)))^(10 sin x) find solution

$$ \\ $$$$ \\ $$$$\left(\mathrm{1}+\mathrm{sin}\:\frac{\pi}{\mathrm{7}}\right)^{\mathrm{3}−\mathrm{cos}\:\mathrm{2x}} =\:\left(\mathrm{sin}\:\frac{\pi}{\mathrm{14}}+\mathrm{cos}\:\frac{\pi}{\mathrm{14}}\right)^{\mathrm{10}\:\mathrm{sin}\:\mathrm{x}} \\ $$$$\mathrm{find}\:\mathrm{solution} \\ $$

Question Number 78770 Answers: 0 Comments: 0

Question Number 78762 Answers: 1 Comments: 1

3acr^2 (1−r)+3apr(1−r)(pa+qb) +3bqr(1−r)(pa+qb) = 3(1−r)^2 (pa+qb)^2 +r^2 b^2 Find p, q, r such that the equation is satisfied for general any values of a,b,c.

$$\mathrm{3}{acr}^{\mathrm{2}} \left(\mathrm{1}−{r}\right)+\mathrm{3}{apr}\left(\mathrm{1}−{r}\right)\left({pa}+{qb}\right) \\ $$$$+\mathrm{3}{bqr}\left(\mathrm{1}−{r}\right)\left({pa}+{qb}\right) \\ $$$$\:\:\:=\:\mathrm{3}\left(\mathrm{1}−{r}\right)^{\mathrm{2}} \left({pa}+{qb}\right)^{\mathrm{2}} +{r}^{\mathrm{2}} {b}^{\mathrm{2}} \\ $$$${Find}\:{p},\:{q},\:{r}\:{such}\:{that}\:{the}\:{equation} \\ $$$${is}\:{satisfied}\:{for}\:{general}\:{any} \\ $$$${values}\:{of}\:{a},{b},{c}.\: \\ $$

Question Number 78755 Answers: 0 Comments: 2

Solve the equation: xy + 5x + 5y = − 25 ... (i) yz + 3y + 5z = − 15 ... (ii) xz + 5z + 3x = − 15 ... (iii)

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\:\:\:\:\mathrm{xy}\:+\:\mathrm{5x}\:+\:\mathrm{5y}\:\:=\:\:−\:\mathrm{25}\:\:\:\:\:\:...\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\:\mathrm{yz}\:+\:\mathrm{3y}\:+\:\mathrm{5z}\:\:=\:\:−\:\mathrm{15}\:\:\:\:\:\:...\:\left(\mathrm{ii}\right) \\ $$$$\:\:\:\:\mathrm{xz}\:+\:\mathrm{5z}\:+\:\mathrm{3x}\:\:=\:\:−\:\mathrm{15}\:\:\:\:\:\:...\:\left(\mathrm{iii}\right) \\ $$

Question Number 78766 Answers: 1 Comments: 0

∫2 e^(1/(2(x−2)^2 )) dx

$$\int\mathrm{2}\:{e}^{\frac{\mathrm{1}}{\mathrm{2}\left({x}−\mathrm{2}\right)^{\mathrm{2}} }} \:{dx} \\ $$

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