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AlgebraQuestion and Answers: Page 1

Question Number 227891 Answers: 1 Comments: 0

Question Number 227880 Answers: 1 Comments: 0

Find: ((log_2 ^2 20 − log_2 ^2 5)/(log_2 10)) = ?

$$\mathrm{Find}:\:\:\:\frac{\mathrm{log}_{\mathrm{2}} ^{\mathrm{2}} \:\mathrm{20}\:−\:\mathrm{log}_{\mathrm{2}} ^{\mathrm{2}} \:\mathrm{5}}{\mathrm{log}_{\mathrm{2}} \mathrm{10}}\:=\:? \\ $$

Question Number 227878 Answers: 0 Comments: 4

Find: 5^((log_5 3)^(144) ) = ?

$$\mathrm{Find}: \\ $$$$\mathrm{5}^{\left(\boldsymbol{\mathrm{log}}_{\mathrm{5}} \mathrm{3}\right)^{\mathrm{144}} } \:\:=\:\:? \\ $$

Question Number 227853 Answers: 1 Comments: 0

prove:Σ_(i=1) ^n (ln((i+1)/i))^2 <(n/(n+1))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{prove}:\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\mathrm{ln}\frac{{i}+\mathrm{1}}{{i}}\right)^{\mathrm{2}} <\frac{{n}}{{n}+\mathrm{1}} \\ $$

Question Number 227862 Answers: 2 Comments: 0

if a_(n+1) =(3/(4−a_n )) for n≥1 and a_1 =0, find a_n in terms of n.

$${if}\:{a}_{{n}+\mathrm{1}} =\frac{\mathrm{3}}{\mathrm{4}−{a}_{{n}} }\:{for}\:{n}\geqslant\mathrm{1}\:{and}\:{a}_{\mathrm{1}} =\mathrm{0}, \\ $$$${find}\:{a}_{{n}} \:{in}\:{terms}\:{of}\:{n}. \\ $$

Question Number 227823 Answers: 2 Comments: 1

Question Number 227811 Answers: 1 Comments: 1

Question Number 227805 Answers: 1 Comments: 0

∫_0 ^(+∞) sin(x)sin(x^2 )dx

$$\int_{\mathrm{0}} ^{+\infty} {sin}\left({x}\right){sin}\left({x}^{\mathrm{2}} \right){dx} \\ $$

Question Number 227796 Answers: 2 Comments: 0

Question Number 227792 Answers: 1 Comments: 0

Prove that, ((a/b)+(b/a))^2 +((b/c)+(c/b))^2 +((c/a)+(a/c))^2 −4 =((a/b)+(b/a))((b/c)+(c/b))((c/a)+(a/c))

$$\:\:{Prove}\:{that}, \\ $$$$\:\left(\frac{{a}}{{b}}+\frac{{b}}{{a}}\right)^{\mathrm{2}} +\left(\frac{{b}}{{c}}+\frac{{c}}{{b}}\right)^{\mathrm{2}} +\left(\frac{{c}}{{a}}+\frac{{a}}{{c}}\right)^{\mathrm{2}} −\mathrm{4} \\ $$$$\:=\left(\frac{{a}}{{b}}+\frac{{b}}{{a}}\right)\left(\frac{{b}}{{c}}+\frac{{c}}{{b}}\right)\left(\frac{{c}}{{a}}+\frac{{a}}{{c}}\right) \\ $$

Question Number 227787 Answers: 0 Comments: 3

Question Number 227715 Answers: 3 Comments: 0

x > 6 y > 2 (√(x^2 − 36)) + (√(y^2 − 4)) = 6 min { x + y } = ?

$$\mathrm{x}\:>\:\mathrm{6} \\ $$$$\mathrm{y}\:>\:\mathrm{2} \\ $$$$\sqrt{\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{36}}\:\:+\:\:\sqrt{\mathrm{y}^{\mathrm{2}} \:−\:\mathrm{4}}\:\:=\:\:\mathrm{6} \\ $$$$\mathrm{min}\:\left\{\:\mathrm{x}\:+\:\mathrm{y}\:\right\}\:=\:? \\ $$

Question Number 227772 Answers: 2 Comments: 0

if (x+(√(x^2 +4)))(y+(√(y^2 +4)))=4, find the minimum of (x^2 +2y)=?

$${if}\:\left({x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{4}}\right)\left({y}+\sqrt{{y}^{\mathrm{2}} +\mathrm{4}}\right)=\mathrm{4},\:{find} \\ $$$${the}\:{minimum}\:{of}\:\left({x}^{\mathrm{2}} +\mathrm{2}{y}\right)=? \\ $$

Question Number 227677 Answers: 1 Comments: 0

z = 2−3i (1−z) ∙ ((1 + z^2 )/4) = ?

$$\mathrm{z}\:=\:\mathrm{2}−\mathrm{3i} \\ $$$$\left(\mathrm{1}−\mathrm{z}\right)\:\centerdot\:\frac{\mathrm{1}\:+\:\mathrm{z}^{\mathrm{2}} }{\mathrm{4}}\:=\:? \\ $$

Question Number 227667 Answers: 1 Comments: 0

sin7xcos5x−cos7xsin5x = (1/2) x = ?

$$\mathrm{sin7xcos5x}−\mathrm{cos7xsin5x}\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{x}\:=\:? \\ $$

Question Number 227651 Answers: 1 Comments: 0

log_5 x = 25 x = ?

$$\mathrm{log}_{\mathrm{5}} \:\mathrm{x}\:=\:\mathrm{25} \\ $$$$\mathrm{x}\:=\:? \\ $$

Question Number 227649 Answers: 1 Comments: 0

25^(log_5 (√2) + 1) = ?

$$\mathrm{25}^{\boldsymbol{\mathrm{log}}_{\mathrm{5}} \:\sqrt{\mathrm{2}}\:+\:\mathrm{1}} \:\:\:=\:\:\:? \\ $$

Question Number 227634 Answers: 0 Comments: 0

prove: p>1,n≥2 (1/n^p )<(1/(p−1))∙[(1/((n−1)^(p− 1) ))−(1/n^(p−1) )]

$$\mathrm{prove}:\:\:\:\:\:\:\:\:\:{p}>\mathrm{1},{n}\geqslant\mathrm{2} \\ $$$$\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{{n}^{{p}} }<\frac{\mathrm{1}}{{p}−\mathrm{1}}\centerdot\left[\frac{\mathrm{1}}{\left({n}−\mathrm{1}\right)^{{p}−\:\mathrm{1}} }−\frac{\mathrm{1}}{{n}^{{p}−\mathrm{1}} }\right] \\ $$

Question Number 227633 Answers: 1 Comments: 0

(3/(log_2 23!)) + (3/(log_3 23!)) + (3/(log_4 23!)) +...+ (3/(log_(22) 23!)) + (3/(log_(23) 23!)) = ?

$$\frac{\mathrm{3}}{\mathrm{log}_{\mathrm{2}} \mathrm{23}!}\:+\:\frac{\mathrm{3}}{\mathrm{log}_{\mathrm{3}} \mathrm{23}!}\:+\:\frac{\mathrm{3}}{\mathrm{log}_{\mathrm{4}} \mathrm{23}!}\:+...+\:\frac{\mathrm{3}}{\mathrm{log}_{\mathrm{22}} \mathrm{23}!}\:+\:\frac{\mathrm{3}}{\mathrm{log}_{\mathrm{23}} \mathrm{23}!}\:=\:? \\ $$

Question Number 227632 Answers: 1 Comments: 0

If 4a^2 + 9b^2 = 13ab Find ((2lg(2a + 3b)−lg25)/(5lg(ab))) = ?

$$\mathrm{If}\:\:\:\mathrm{4a}^{\mathrm{2}} \:+\:\mathrm{9b}^{\mathrm{2}} \:=\:\mathrm{13ab} \\ $$$$\mathrm{Find}\:\:\:\frac{\mathrm{2}\boldsymbol{\mathrm{lg}}\left(\mathrm{2a}\:+\:\mathrm{3b}\right)−\boldsymbol{\mathrm{lg}}\mathrm{25}}{\mathrm{5}\boldsymbol{\mathrm{lg}}\left(\mathrm{ab}\right)}\:=\:? \\ $$

Question Number 227582 Answers: 1 Comments: 0

cos^2 (((3x)/2) + ((11π)/2)) - sin^2 (((3x)/2) + ((11π)/2)) = - (1/2) x = ?

$$\mathrm{cos}^{\mathrm{2}} \left(\frac{\mathrm{3x}}{\mathrm{2}}\:+\:\frac{\mathrm{11}\pi}{\mathrm{2}}\right)\:-\:\mathrm{sin}^{\mathrm{2}} \left(\frac{\mathrm{3x}}{\mathrm{2}}\:+\:\frac{\mathrm{11}\pi}{\mathrm{2}}\right)\:=\:-\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{x}\:=\:? \\ $$

Question Number 227580 Answers: 1 Comments: 0

(π/4)arctan(tan((7π)/8))+tan2x=(1/2)cos(arccos(−(1/2))+(π/3)) x = ?

$$\frac{\pi}{\mathrm{4}}\mathrm{arctan}\left(\mathrm{tan}\frac{\mathrm{7}\pi}{\mathrm{8}}\right)+\mathrm{tan2x}=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\left(\mathrm{arccos}\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)+\frac{\pi}{\mathrm{3}}\right) \\ $$$$\mathrm{x}\:=\:? \\ $$

Question Number 227579 Answers: 1 Comments: 0

cos6x = ((2tan(π/8))/(1−tan^2 (π/8))) ⇒ x = ?

$$\mathrm{cos6x}\:=\:\frac{\mathrm{2tan}\frac{\pi}{\mathrm{8}}}{\mathrm{1}−\mathrm{tan}^{\mathrm{2}} \frac{\pi}{\mathrm{8}}}\:\:\:\:\:\Rightarrow\:\:\:\:\mathrm{x}\:=\:? \\ $$

Question Number 227578 Answers: 1 Comments: 0

sin3x + sin^2 (π/4) = cos^2 (π/4) + ((√3)/2) x = ?

$$\mathrm{sin3x}\:+\:\mathrm{sin}^{\mathrm{2}} \frac{\pi}{\mathrm{4}}\:=\:\mathrm{cos}^{\mathrm{2}} \frac{\pi}{\mathrm{4}}\:+\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}} \\ $$$$\mathrm{x}\:=\:? \\ $$

Question Number 227489 Answers: 1 Comments: 1

1+2−3−4+5+6−7−8+9+10−11−12+ ... +100 =?

$$\mathrm{1}+\mathrm{2}−\mathrm{3}−\mathrm{4}+\mathrm{5}+\mathrm{6}−\mathrm{7}−\mathrm{8}+\mathrm{9}+\mathrm{10}−\mathrm{11}−\mathrm{12}+ \\ $$$$...\:+\mathrm{100}\:=? \\ $$

Question Number 227476 Answers: 3 Comments: 0

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