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AlgebraQuestion and Answers: Page 63

Question Number 192057 Answers: 1 Comments: 0

Simplify: (((√a) + (√a^2 ) + (√a^3 ) + (√a^4 ))/(((√a) + 1)∙(a + 1)))

$$\mathrm{Simplify}: \\ $$$$\frac{\sqrt{\mathrm{a}}\:\:+\:\:\sqrt{\mathrm{a}^{\mathrm{2}} }\:\:+\:\:\sqrt{\mathrm{a}^{\mathrm{3}} }\:\:+\:\:\sqrt{\mathrm{a}^{\mathrm{4}} }}{\left(\sqrt{\mathrm{a}}\:\:+\:\:\mathrm{1}\right)\centerdot\left(\mathrm{a}\:\:+\:\:\mathrm{1}\right)} \\ $$

Question Number 192039 Answers: 0 Comments: 0

let x,y,z be positive real number such that x^4 +y^4 +z^4 = 1 find the minimum value of (x^3 /(1−x^8 )) + (y^3 /(1−y^8 )) + (z^3 /(1−z^8 ))

$$\mathrm{let}\:{x},{y},{z}\:\mathrm{be}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that} \\ $$$$\:{x}^{\mathrm{4}} +{y}^{\mathrm{4}} +{z}^{\mathrm{4}} \:=\:\mathrm{1}\:\mathrm{find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value} \\ $$$$\mathrm{of} \\ $$$$\:\:\:\:\frac{{x}^{\mathrm{3}} }{\mathrm{1}−{x}^{\mathrm{8}} }\:+\:\frac{{y}^{\mathrm{3}} }{\mathrm{1}−{y}^{\mathrm{8}} }\:+\:\frac{{z}^{\mathrm{3}} }{\mathrm{1}−{z}^{\mathrm{8}} } \\ $$

Question Number 192009 Answers: 2 Comments: 0

if a>1 , show ((Σ_(k=1) ^(a^2 −1) (√(a+(√k))))/(Σ_(k=1) ^(a^2 −1) (√(a−(√k))))) = (√2) + 1

$$\:\:\:\:\:{if}\:\:{a}>\mathrm{1}\:,\:{show} \\ $$$$\:\:\:\:\:\frac{\underset{{k}=\mathrm{1}} {\overset{{a}^{\mathrm{2}} −\mathrm{1}} {\sum}}\:\:\sqrt{{a}+\sqrt{{k}}}}{\underset{{k}=\mathrm{1}} {\overset{{a}^{\mathrm{2}} −\mathrm{1}} {\sum}}\:\:\sqrt{{a}−\sqrt{{k}}}}\:\:\:=\:\:\:\sqrt{\mathrm{2}}\:\:+\:\:\mathrm{1} \\ $$

Question Number 191961 Answers: 0 Comments: 0

Question Number 191950 Answers: 1 Comments: 0

(√a) = 1 + (1/a) find: a^2 −a−(√a) = ?

$$\sqrt{\mathrm{a}}\:=\:\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{a}}\:\:\:\mathrm{find}:\:\:\:\mathrm{a}^{\mathrm{2}} −\mathrm{a}−\sqrt{\mathrm{a}}\:=\:? \\ $$

Question Number 191946 Answers: 1 Comments: 0

((fof^(−1) (5)+fof^(−1) (15))/(fof^(−1) (5)))=?

$$\frac{\mathrm{fof}^{−\mathrm{1}} \left(\mathrm{5}\right)+\mathrm{fof}^{−\mathrm{1}} \left(\mathrm{15}\right)}{\mathrm{fof}^{−\mathrm{1}} \left(\mathrm{5}\right)}=? \\ $$

Question Number 191947 Answers: 2 Comments: 0

prove that (√(a+b(√(a−b(√(a+b(√(a−b(√(...)))))))))) = (((√(4a−3b^2 ))+b)/2)

$${prove}\:{that} \\ $$$$\sqrt{{a}+{b}\sqrt{{a}−{b}\sqrt{{a}+{b}\sqrt{{a}−{b}\sqrt{...}}}}}\:\:=\:\:\frac{\sqrt{\mathrm{4}{a}−\mathrm{3}{b}^{\mathrm{2}} }+{b}}{\mathrm{2}} \\ $$

Question Number 191867 Answers: 1 Comments: 0

Prove that if u=f(x^3 +y^3 ),where f is arbitry function then x^2 (∂u/∂y) = y^2 (∂u/∂x)

$${Prove}\:{that}\:{if}\:\:\:{u}={f}\left({x}^{\mathrm{3}} +{y}^{\mathrm{3}} \right),{where}\:{f}\:\:{is}\:{arbitry} \\ $$$${function}\:{then}\:\:\:\:{x}^{\mathrm{2}} \:\frac{\partial{u}}{\partial{y}}\:=\:{y}^{\mathrm{2}} \frac{\partial{u}}{\partial{x}} \\ $$

Question Number 191839 Answers: 1 Comments: 0

2^a = 3^b = 36^c then prove that ab = 2c(a + b).

$$\mathrm{2}^{{a}} \:=\:\mathrm{3}^{{b}} \:=\:\mathrm{36}^{{c}} \:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$${ab}\:=\:\mathrm{2}{c}\left({a}\:+\:{b}\right). \\ $$

Question Number 191833 Answers: 3 Comments: 0

Question Number 191830 Answers: 0 Comments: 0

Question Number 191798 Answers: 2 Comments: 2

Question Number 191790 Answers: 1 Comments: 7

prove that ((2x−4)/(2∙3∙4))+((3x−5)/(3∙4∙5))+((4x−6)/(4∙5∙6))+.....+((100x−102)/(100∙101∙102))=((103)/(102))

$${prove}\:{that} \\ $$$$\frac{\mathrm{2}{x}−\mathrm{4}}{\mathrm{2}\centerdot\mathrm{3}\centerdot\mathrm{4}}+\frac{\mathrm{3}{x}−\mathrm{5}}{\mathrm{3}\centerdot\mathrm{4}\centerdot\mathrm{5}}+\frac{\mathrm{4}{x}−\mathrm{6}}{\mathrm{4}\centerdot\mathrm{5}\centerdot\mathrm{6}}+.....+\frac{\mathrm{100}{x}−\mathrm{102}}{\mathrm{100}\centerdot\mathrm{101}\centerdot\mathrm{102}}=\frac{\mathrm{103}}{\mathrm{102}} \\ $$$$ \\ $$

Question Number 191775 Answers: 2 Comments: 0

Question Number 191735 Answers: 3 Comments: 0

Verify that ┐(p→q)→(p∧^┐ q) is tautology using laws of algebra

$${Verify}\:{that} \\ $$$$\:\urcorner\left({p}\rightarrow{q}\right)\rightarrow\left({p}\wedge^{\urcorner} {q}\right)\:{is}\:{tautology}\:{using}\:{laws}\:{of} \\ $$$${algebra} \\ $$

Question Number 191753 Answers: 1 Comments: 0

Question Number 191717 Answers: 0 Comments: 0

Question Number 191716 Answers: 0 Comments: 0

Question Number 191706 Answers: 2 Comments: 0

Question Number 191675 Answers: 2 Comments: 2

Solve for x : (x − (1/x))^(1/2) + (1 − (1/x))^(1/2) = x

$$\mathrm{Solve}\:\mathrm{for}\:{x}\:: \\ $$$$\left({x}\:−\:\frac{\mathrm{1}}{{x}}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \:+\:\left(\mathrm{1}\:−\:\frac{\mathrm{1}}{{x}}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \:=\:{x} \\ $$

Question Number 191663 Answers: 4 Comments: 0

2^a +4^b +8^c =328 find a,b and c when(a,b,c)is natual number

$$\mathrm{2}^{{a}} +\mathrm{4}^{{b}} +\mathrm{8}^{{c}} =\mathrm{328} \\ $$$${find}\:{a},{b}\:{and}\:{c} \\ $$$${when}\left({a},{b},{c}\right){is}\:{natual}\:{number} \\ $$

Question Number 191632 Answers: 0 Comments: 1

∫_0 ^∞ (√(tan θ)) dθ

$$\int_{\mathrm{0}} ^{\infty} \sqrt{\mathrm{tan}\:\theta}\:{d}\theta \\ $$

Question Number 191631 Answers: 1 Comments: 0

∫_0 ^∞ x^(1/2) e^(−x^2 ) dx

$$\int_{\mathrm{0}} ^{\infty} {x}^{\frac{\mathrm{1}}{\mathrm{2}}} {e}^{−{x}^{\mathrm{2}} } {dx} \\ $$

Question Number 191624 Answers: 1 Comments: 0

Question Number 191623 Answers: 2 Comments: 0

Question Number 191615 Answers: 1 Comments: 0

a + b + c = 0. Prove that, (a/(a^2 − bc)) + (b/(b^2 − ca)) + (c/(c^2 − ab)) = 0.

$${a}\:+\:{b}\:+\:{c}\:=\:\mathrm{0}.\:\mathrm{Prove}\:\mathrm{that}, \\ $$$$\frac{{a}}{{a}^{\mathrm{2}} \:−\:{bc}}\:+\:\frac{{b}}{{b}^{\mathrm{2}} \:−\:{ca}}\:+\:\frac{{c}}{{c}^{\mathrm{2}} \:−\:{ab}}\:=\:\mathrm{0}. \\ $$

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