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AlgebraQuestion and Answers: Page 1

Question Number 217872 Answers: 0 Comments: 0

a , b , c > 0 ab + ac + bc = 1 Prove that: (√(a^3 + a)) + (√(b^3 + b)) + (√(c^3 + c)) ≥ 2 (√(a + b + c))

$$\mathrm{a}\:,\:\mathrm{b}\:,\:\mathrm{c}\:>\:\mathrm{0} \\ $$$$\mathrm{ab}\:+\:\mathrm{ac}\:+\:\mathrm{bc}\:=\:\mathrm{1} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\sqrt{\mathrm{a}^{\mathrm{3}} \:+\:\mathrm{a}}\:\:+\:\:\sqrt{\mathrm{b}^{\mathrm{3}} \:+\:\mathrm{b}}\:\:+\:\:\sqrt{\mathrm{c}^{\mathrm{3}} \:+\:\mathrm{c}}\:\:\geqslant\:\mathrm{2}\:\sqrt{\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}} \\ $$

Question Number 217871 Answers: 0 Comments: 0

n ≥ 2 Prove that: Π_(k=1) ^n tg [ (π/3) (1 + (3^k /(3^n − 1)))] = Π_(k=1) ^n ctg [ (π/3) (1 − (3^k /(3^n − 1)))]

Question Number 217858 Answers: 0 Comments: 0

Question Number 217857 Answers: 1 Comments: 0

Prove that: ((cos20°))^(1/3) + ((cos80°))^(1/3) + ((cos160°))^(1/3) = (((3 ∙ (9)^(1/3) − 6)/2))^(1/3)

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{cos20}°}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{cos80}°}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{cos160}°}\:=\:\sqrt[{\mathrm{3}}]{\frac{\mathrm{3}\:\centerdot\:\sqrt[{\mathrm{3}}]{\mathrm{9}}\:−\:\mathrm{6}}{\mathrm{2}}} \\ $$

Question Number 217849 Answers: 0 Comments: 0

Question Number 217843 Answers: 0 Comments: 5

Question Number 217822 Answers: 1 Comments: 0

Question Number 217802 Answers: 1 Comments: 1

Question Number 217789 Answers: 0 Comments: 4

Question Number 217766 Answers: 3 Comments: 0

Solve for x & y (1/x)+(1/y)=5 (1/x^2 )+(1/y^2 )=13

$${Solve}\:{for}\:{x}\:\&\:{y} \\ $$$$\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}=\mathrm{5} \\ $$$$\frac{\mathrm{1}}{{x}^{\mathrm{2}} }+\frac{\mathrm{1}}{{y}^{\mathrm{2}} }=\mathrm{13} \\ $$

Question Number 217764 Answers: 1 Comments: 0

Let a, b, c be distinct real numbers such that (a/(b−c))+(b/(c−a))+(c/(a−b))=0 then prove that (a/((b−c)^2 ))+(b/((c−a)^2 ))+(c/((a−b)^2 ))=0

$$ \\ $$$$\mathrm{Let}\:\mathrm{a},\:\mathrm{b},\:\mathrm{c}\:\mathrm{be}\:\mathrm{distinct}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that} \\ $$$$\frac{{a}}{{b}−{c}}+\frac{{b}}{{c}−{a}}+\frac{{c}}{{a}−{b}}=\mathrm{0} \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{{a}}{\left({b}−{c}\right)^{\mathrm{2}} }+\frac{{b}}{\left({c}−{a}\right)^{\mathrm{2}} }+\frac{{c}}{\left({a}−{b}\right)^{\mathrm{2}} }=\mathrm{0} \\ $$

Question Number 217735 Answers: 1 Comments: 1

a + b + c= abc a^2 + b^2 + c^2 = 49 ab+bc+ca=? . .

$$\:\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}=\:\mathrm{abc} \\ $$$$\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:+\:\mathrm{c}^{\mathrm{2}} \:=\:\mathrm{49}\: \\ $$$$\mathrm{ab}+\mathrm{bc}+\mathrm{ca}=?\:\:\:.\:\:\:\:\:\:\:\:\:\:\:.\:\:\:\:\:\:\:\: \\ $$

Question Number 217733 Answers: 3 Comments: 0

Question Number 217732 Answers: 2 Comments: 1

{ ((x+y=xy)),((x^2 +y^2 =25)) :}; x^4 +y^4 =?

$$\begin{cases}{{x}+{y}={xy}}\\{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{25}}\end{cases};\:\:{x}^{\mathrm{4}} +{y}^{\mathrm{4}} =? \\ $$

Question Number 217730 Answers: 3 Comments: 0

x+(1/x)=3 ; x^5 +(1/x^5 )=?

$${x}+\frac{\mathrm{1}}{{x}}=\mathrm{3}\:;\:{x}^{\mathrm{5}} +\frac{\mathrm{1}}{{x}^{\mathrm{5}} }=? \\ $$

Question Number 217690 Answers: 1 Comments: 3

Question Number 217664 Answers: 1 Comments: 0

If f(x) = (√(2x + 3)) prove that: f ′(x) = (1/( (√(2x + 3))))

$$\mathrm{If}\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\sqrt{\mathrm{2x}\:+\:\mathrm{3}} \\ $$$$\mathrm{prove}\:\mathrm{that}:\:\:\:\mathrm{f}\:'\left(\mathrm{x}\right)\:=\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2x}\:+\:\mathrm{3}}}\: \\ $$

Question Number 217660 Answers: 1 Comments: 0

f(x) + f(y)=f(x+y)+xy f(x)=?

$$\:{f}\left({x}\right)\:+\:{f}\left({y}\right)={f}\left({x}+{y}\right)+{xy}\: \\ $$$${f}\left({x}\right)=? \\ $$

Question Number 217659 Answers: 3 Comments: 0

x+y=7 ∧ x^3 +y^3 =133; x,y=?

$${x}+{y}=\mathrm{7}\:\wedge\:{x}^{\mathrm{3}} +{y}^{\mathrm{3}} =\mathrm{133};\:{x},{y}=? \\ $$

Question Number 217643 Answers: 2 Comments: 0

a^(1/3) +b^(1/3) +c^(1/3) =(1/(3^(1/3) −2^(1/3) )) b+c−a=?

$$\:{a}^{\frac{\mathrm{1}}{\mathrm{3}}} +{b}^{\frac{\mathrm{1}}{\mathrm{3}}} +{c}^{\frac{\mathrm{1}}{\mathrm{3}}} =\frac{\mathrm{1}}{\mathrm{3}^{\frac{\mathrm{1}}{\mathrm{3}}} −\mathrm{2}^{\frac{\mathrm{1}}{\mathrm{3}}} } \\ $$$$\:\:{b}+{c}−{a}=? \\ $$

Question Number 217579 Answers: 0 Comments: 4

f : R → R f(f(x)) = x^2 − x + 1 f(x) = ? Altered Question# 217541

$$\mathrm{f}\::\:\mathbb{R}\:\rightarrow\:\mathbb{R} \\ $$$$\mathrm{f}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\:=\:\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{x}\:+\:\mathrm{1} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\:=\:? \\ $$$$\mathrm{Altered}\:\mathrm{Question}#\:\mathrm{217541} \\ $$

Question Number 217541 Answers: 1 Comments: 0

f : R → R f(f(x)) = x^2 − x + 1 f(0) = ?

Question Number 217535 Answers: 2 Comments: 0

A two-digit number is such that the sum of its digits is 10. When the digits are reversed, the new number is 28 less than twice the original number. Find the original number.

$$\mathrm{A}\:\mathrm{two}-\mathrm{digit}\:\mathrm{number}\:\mathrm{is}\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{its}\:\mathrm{digits}\:\mathrm{is}\:\mathrm{10}.\:\mathrm{When} \\ $$$$\mathrm{the}\:\mathrm{digits}\:\mathrm{are}\:\mathrm{reversed},\:\mathrm{the}\:\mathrm{new}\:\mathrm{number} \\ $$$$\:\mathrm{is}\:\mathrm{28}\:\mathrm{less}\:\mathrm{than}\:\mathrm{twice}\:\mathrm{the}\:\:\mathrm{original}\:\mathrm{number}. \\ $$$$\:\mathrm{Find}\:\mathrm{the}\:\mathrm{original}\:\mathrm{number}. \\ $$

Question Number 217482 Answers: 0 Comments: 0

( n(√((n+1))) +(n+1)(√n) )^n +( n(√((n+1))) −(n+1)(√n) )^n =n^2 (n+1)(n+3)

$$\left(\:{n}\sqrt{\left({n}+\mathrm{1}\right)}\:+\left({n}+\mathrm{1}\right)\sqrt{{n}}\:\:\right)^{{n}} +\left(\:{n}\sqrt{\left({n}+\mathrm{1}\right)}\:−\left({n}+\mathrm{1}\right)\sqrt{{n}}\:\:\right)^{{n}} ={n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)\left({n}+\mathrm{3}\right) \\ $$

Question Number 217448 Answers: 3 Comments: 2

Solve for x: (1+(√2) )^x +(1−(√2) )^x =14

$${Solve}\:{for}\:{x}: \\ $$$$\left(\mathrm{1}+\sqrt{\mathrm{2}}\:\right)^{{x}} +\left(\mathrm{1}−\sqrt{\mathrm{2}}\:\right)^{{x}} =\mathrm{14} \\ $$

Question Number 217424 Answers: 2 Comments: 0

The angle of elevation of the top of a building 24 m high is observed from the top and from the bottom of a vertical ladder, and found to be 45° and 60° respectively. Find the height of the ladder.

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