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AlgebraQuestion and Answers: Page 269

Question Number 89070 Answers: 0 Comments: 0

(((√(a^2 +(b^2 /x^2 )))+(√(a^2 +b^2 x^2 )))/((√(a^2 +(b^2 /x^2 )))−(√(a^2 +b^2 x^2 )))) = ((a^2 /b^2 ))x x∈R^+ ; Express x in terms of a^2 , b^2 .

$$\frac{\sqrt{{a}^{\mathrm{2}} +\frac{{b}^{\mathrm{2}} }{{x}^{\mathrm{2}} }}+\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} {x}^{\mathrm{2}} }}{\sqrt{{a}^{\mathrm{2}} +\frac{{b}^{\mathrm{2}} }{{x}^{\mathrm{2}} }}−\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} {x}^{\mathrm{2}} }}\:=\:\left(\frac{{a}^{\mathrm{2}} }{{b}^{\mathrm{2}} }\right){x} \\ $$$${x}\in\mathbb{R}^{+} \:\:;\:\:{Express}\:{x}\:{in}\:{terms}\:{of} \\ $$$${a}^{\mathrm{2}} ,\:{b}^{\mathrm{2}} \:. \\ $$

Question Number 89032 Answers: 0 Comments: 0

find by using de moivre′s formula cos(2°)=?

$${find}\:{by}\:{using}\:{de}\:{moivre}'{s}\:{formula} \\ $$$${cos}\left(\mathrm{2}°\right)=? \\ $$

Question Number 88937 Answers: 0 Comments: 0

Question Number 88921 Answers: 2 Comments: 0

find x,y x−2y−(√(xy))=0 (√(x−1))−(√(2y−1))=1

$${find}\:{x},{y} \\ $$$${x}−\mathrm{2}{y}−\sqrt{{xy}}=\mathrm{0} \\ $$$$\sqrt{{x}−\mathrm{1}}−\sqrt{\mathrm{2}{y}−\mathrm{1}}=\mathrm{1} \\ $$

Question Number 88881 Answers: 1 Comments: 0

Question Number 88873 Answers: 0 Comments: 1

3+(√(x^2 −5)) > ∣x−1∣

$$\mathrm{3}+\sqrt{{x}^{\mathrm{2}} −\mathrm{5}}\:>\:\mid{x}−\mathrm{1}\mid\: \\ $$

Question Number 88858 Answers: 0 Comments: 0

Question Number 88811 Answers: 1 Comments: 0

{ (((x+1)^2 (y+1)^2 =27xy)),(((x^2 +1)(y^2 +1) =10xy)) :}

$$\begin{cases}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} \left({y}+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{27}{xy}}\\{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({y}^{\mathrm{2}} +\mathrm{1}\right)\:=\mathrm{10}{xy}}\end{cases} \\ $$

Question Number 88771 Answers: 0 Comments: 4

solve x^x^4 =64

$${solve} \\ $$$${x}^{{x}^{\mathrm{4}} } =\mathrm{64} \\ $$

Question Number 88752 Answers: 1 Comments: 2

cos(𝛂)+cos(𝛃)+cos(𝛄)≤(3/2) prove the inequality

$$\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\alpha}\right)+\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\beta}\right)+\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\gamma}\right)\leqslant\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{prove}}\:\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{inequality}} \\ $$

Question Number 88678 Answers: 0 Comments: 8

Find (√i)+(√(−i))

$$\boldsymbol{\mathrm{F}}{ind}\:\:\:\sqrt{\boldsymbol{{i}}}+\sqrt{−\boldsymbol{\mathrm{i}}} \\ $$

Question Number 88642 Answers: 1 Comments: 2

Question Number 88611 Answers: 0 Comments: 0

prove that ((1+p^2 +p^4 +......+p^(2n) )/(p+p^3 +p^5 +.....p^(2n−1) ))>((n+1)/(np))

$${prove}\:{that} \\ $$$$\frac{\mathrm{1}+{p}^{\mathrm{2}} +{p}^{\mathrm{4}} +......+{p}^{\mathrm{2}{n}} }{{p}+{p}^{\mathrm{3}} +{p}^{\mathrm{5}} +.....{p}^{\mathrm{2}{n}−\mathrm{1}} }>\frac{{n}+\mathrm{1}}{{np}} \\ $$

Question Number 88610 Answers: 0 Comments: 2

is 1×1×1×..........=1^∞ or 1×1×1×1×1×..........=1

$${is}\:\mathrm{1}×\mathrm{1}×\mathrm{1}×..........=\mathrm{1}^{\infty} \: \\ $$$${or} \\ $$$$\:\mathrm{1}×\mathrm{1}×\mathrm{1}×\mathrm{1}×\mathrm{1}×..........=\mathrm{1} \\ $$$$ \\ $$$$ \\ $$

Question Number 88594 Answers: 0 Comments: 3

solve for x∈C cos (x)=a+bi

$${solve}\:{for}\:{x}\in\mathbb{C} \\ $$$$\mathrm{cos}\:\left({x}\right)={a}+{bi} \\ $$

Question Number 88590 Answers: 0 Comments: 0

a^a^a^a^3 =5 find−a

$${a}^{{a}^{{a}^{{a}^{\mathrm{3}} } } } =\mathrm{5} \\ $$$${find}−{a} \\ $$

Question Number 88491 Answers: 1 Comments: 0

solve cos(x)=k

$$\boldsymbol{{solve}} \\ $$$${cos}\left({x}\right)={k} \\ $$

Question Number 88458 Answers: 0 Comments: 0

Using the principle of mathematical induction to prove that a_1 , a_2 , ... , a_n , ((a_1 + a_2 + ... + a_n )/n) ≥ ((a_1 , a_2 , ... , a_n ))^(1/n)

$$\mathrm{Using}\:\mathrm{the}\:\mathrm{principle}\:\mathrm{of}\:\mathrm{mathematical}\:\mathrm{induction}\:\mathrm{to}\:\mathrm{prove} \\ $$$$\mathrm{that}\:\:\:\mathrm{a}_{\mathrm{1}} \:,\:\:\:\mathrm{a}_{\mathrm{2}} \:,\:\:...\:,\:\mathrm{a}_{\mathrm{n}} \:,\:\:\frac{\mathrm{a}_{\mathrm{1}} \:+\:\mathrm{a}_{\mathrm{2}} \:+\:...\:+\:\mathrm{a}_{\mathrm{n}} }{\mathrm{n}}\:\:\:\:\geqslant\:\:\:\sqrt[{\mathrm{n}}]{\mathrm{a}_{\mathrm{1}} \:,\:\:\mathrm{a}_{\mathrm{2}} \:,\:\:...\:,\:\mathrm{a}_{\mathrm{n}} } \\ $$

Question Number 88435 Answers: 0 Comments: 0

Question Number 88372 Answers: 0 Comments: 10

There are four boxes, each of them contains exactly the same numbers: 1,2,3,...,n. Four different numbers are drawn from the boxes and multiplicated with each other to get a product. What′s the sum of all products? Σ_(a≠b≠c≠d) abcd=?

$${There}\:{are}\:{four}\:{boxes},\:{each}\:{of}\:{them} \\ $$$${contains}\:{exactly}\:{the}\:{same}\:{numbers}: \\ $$$$\mathrm{1},\mathrm{2},\mathrm{3},...,{n}. \\ $$$${Four}\:{different}\:{numbers}\:{are}\:{drawn} \\ $$$${from}\:{the}\:{boxes}\:{and}\:{multiplicated} \\ $$$${with}\:{each}\:{other}\:{to}\:{get}\:{a}\:{product}. \\ $$$${What}'{s}\:{the}\:{sum}\:{of}\:{all}\:{products}? \\ $$$$\underset{{a}\neq{b}\neq{c}\neq{d}} {\sum}{abcd}=? \\ $$

Question Number 88360 Answers: 0 Comments: 2

(e/(√e)) × ((e)^(1/(3 )) /(e)^(1/(4 )) ) × ((e)^(1/(5 )) /(e)^(1/(6 )) ) × ((e)^(1/(7 )) /(e)^(1/(8 )) )×...=?

$$\frac{\mathrm{e}}{\sqrt{\mathrm{e}}}\:×\:\frac{\sqrt[{\mathrm{3}\:\:}]{\mathrm{e}}}{\sqrt[{\mathrm{4}\:\:}]{\mathrm{e}}}\:×\:\frac{\sqrt[{\mathrm{5}\:\:}]{\mathrm{e}}}{\sqrt[{\mathrm{6}\:\:}]{\mathrm{e}}}\:×\:\frac{\sqrt[{\mathrm{7}\:\:}]{\mathrm{e}}}{\sqrt[{\mathrm{8}\:\:}]{\mathrm{e}}}×...=? \\ $$

Question Number 88329 Answers: 0 Comments: 4

Question Number 88314 Answers: 2 Comments: 1

( a,b )are complex numbers and a^2 +ab+b^2 =0 find ((a/(a+b)))^(2020) +((b/(a+b)))^(2020)

$$\left(\:{a},{b}\:\right){are}\:{complex}\:{numbers}\:{and}\:{a}^{\mathrm{2}} +{ab}+{b}^{\mathrm{2}} =\mathrm{0} \\ $$$${find}\:\left(\frac{{a}}{{a}+{b}}\right)^{\mathrm{2020}} +\left(\frac{{b}}{{a}+{b}}\right)^{\mathrm{2020}} \\ $$$$ \\ $$

Question Number 88252 Answers: 0 Comments: 0

Question Number 88251 Answers: 0 Comments: 0

Question Number 88239 Answers: 0 Comments: 0

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