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AlgebraQuestion and Answers: Page 178

Question Number 151549 Answers: 1 Comments: 0

Compare: 2^2^2^.^.^. and 3^3^3^.^.^. Here it is raised 1001 times a square, 1000 times a cube.

$$\mathrm{Compare}: \\ $$$$\mathrm{2}^{\mathrm{2}^{\mathrm{2}^{.^{.^{.} } } } } \:\:\:\:\:\mathrm{and}\:\:\:\:\:\mathrm{3}^{\mathrm{3}^{\mathrm{3}^{.^{.^{.} } } } } \\ $$$$\mathrm{Here}\:\mathrm{it}\:\mathrm{is}\:\mathrm{raised}\:\mathrm{1001}\:\mathrm{times}\:\mathrm{a}\:\mathrm{square}, \\ $$$$\mathrm{1000}\:\mathrm{times}\:\mathrm{a}\:\mathrm{cube}. \\ $$

Question Number 151533 Answers: 0 Comments: 10

Compare: 2^2^2 and 3^3^3

$$\mathrm{Compare}: \\ $$$$\mathrm{2}^{\mathrm{2}^{\mathrm{2}} } \:\:\:\mathrm{and}\:\:\:\mathrm{3}^{\mathrm{3}^{\mathrm{3}} } \\ $$

Question Number 151531 Answers: 0 Comments: 1

Question Number 151490 Answers: 1 Comments: 0

{ ((a^2 =2a+b)),((b^2 =a+2b)) :} and a≠b find (√(a^2 +b^2 +1))

$$\begin{cases}{\mathrm{a}^{\mathrm{2}} =\mathrm{2a}+\mathrm{b}}\\{\mathrm{b}^{\mathrm{2}} =\mathrm{a}+\mathrm{2b}}\end{cases}\:\:\mathrm{and}\:\:\mathrm{a}\neq\mathrm{b}\:\:\mathrm{find}\:\:\sqrt{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{1}} \\ $$

Question Number 151481 Answers: 0 Comments: 2

if x;y;z;t∈R and x+y+z ≤ 3t ; y+z+t ≤ 3x z+t+x ≤ 3y ; t+x+y ≤ 3z compare: x;y;z;t

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y};\mathrm{z};\mathrm{t}\in\mathbb{R}\:\:\mathrm{and} \\ $$$$\mathrm{x}+\mathrm{y}+\mathrm{z}\:\leqslant\:\mathrm{3t}\:\:;\:\:\mathrm{y}+\mathrm{z}+\mathrm{t}\:\leqslant\:\mathrm{3x} \\ $$$$\mathrm{z}+\mathrm{t}+\mathrm{x}\:\leqslant\:\mathrm{3y}\:\:;\:\:\mathrm{t}+\mathrm{x}+\mathrm{y}\:\leqslant\:\mathrm{3z} \\ $$$$\mathrm{compare}:\:\:\mathrm{x};\mathrm{y};\mathrm{z};\mathrm{t} \\ $$

Question Number 151636 Answers: 1 Comments: 4

Question Number 151475 Answers: 2 Comments: 0

if x∈R prove that: x^6 - x + 1 > 0

$$\mathrm{if}\:\:\boldsymbol{\mathrm{x}}\in\mathbb{R}\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\mathrm{x}^{\mathrm{6}} \:-\:\mathrm{x}\:+\:\mathrm{1}\:>\:\mathrm{0} \\ $$

Question Number 151460 Answers: 1 Comments: 0

if f(x) = x^2 - 2x + 2 and g(x) = (√x) + 1 find [ f o g ] (x) = ?

$$\mathrm{if}\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{x}^{\mathrm{2}} \:-\:\mathrm{2x}\:+\:\mathrm{2} \\ $$$$\mathrm{and}\:\:\mathrm{g}\left(\mathrm{x}\right)\:=\:\sqrt{\mathrm{x}}\:+\:\mathrm{1} \\ $$$$\mathrm{find}\:\:\:\left[\:\mathrm{f}\:{o}\:\mathrm{g}\:\right]\:\left(\mathrm{x}\right)\:=\:? \\ $$

Question Number 151453 Answers: 0 Comments: 7

find all continous functions f : R→R such that: f(x^2 + 1) = f((√(1 + x^4 ))) ; ∀x∈R

$$\mathrm{find}\:\mathrm{all}\:\mathrm{continous}\:\mathrm{functions}\:\mathrm{f}\::\:\mathbb{R}\rightarrow\mathbb{R} \\ $$$$\mathrm{such}\:\mathrm{that}: \\ $$$$\mathrm{f}\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{1}\right)\:=\:\mathrm{f}\left(\sqrt{\mathrm{1}\:+\:\mathrm{x}^{\mathrm{4}} }\right)\:;\:\forall\mathrm{x}\in\mathbb{R} \\ $$

Question Number 151449 Answers: 2 Comments: 0

Ω =∫_( 0) ^( ∞) sin(x^2 ) dx = ?

$$\Omega\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\mathrm{sin}\left(\mathrm{x}^{\mathrm{2}} \right)\:\mathrm{dx}\:=\:? \\ $$

Question Number 151448 Answers: 1 Comments: 2

Prove that: Artimetric mean ≥ Geometric mean ((a + b)/2) ≥ (√(ab))

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\boldsymbol{\mathrm{A}}\mathrm{rtimetric}\:\mathrm{mean}\:\geqslant\:\boldsymbol{\mathrm{G}}\mathrm{eometric}\:\mathrm{mean} \\ $$$$\frac{\mathrm{a}\:+\:\mathrm{b}}{\mathrm{2}}\:\geqslant\:\sqrt{\mathrm{ab}} \\ $$

Question Number 151436 Answers: 0 Comments: 0

Question Number 151435 Answers: 1 Comments: 4

Question Number 151418 Answers: 1 Comments: 0

Find the last three digits: 49^(303) ∙ 3993^(202) ∙ 39^(606)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{last}\:\mathrm{three}\:\mathrm{digits}: \\ $$$$\mathrm{49}^{\mathrm{303}} \:\centerdot\:\mathrm{3993}^{\mathrm{202}} \:\centerdot\:\mathrm{39}^{\mathrm{606}} \\ $$

Question Number 151393 Answers: 2 Comments: 0

Question Number 151391 Answers: 1 Comments: 0

if a;b∈R and (a^2 +1)(b^2 +1)+9=6(a+b) find a^2 +b^2 =?

$$\mathrm{if}\:\:\mathrm{a};\mathrm{b}\in\mathbb{R}\:\:\mathrm{and} \\ $$$$\left(\mathrm{a}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{b}^{\mathrm{2}} +\mathrm{1}\right)+\mathrm{9}=\mathrm{6}\left(\mathrm{a}+\mathrm{b}\right) \\ $$$$\mathrm{find}\:\:\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} =? \\ $$

Question Number 151383 Answers: 1 Comments: 1

if x;y>0 peove that: (2/3) ∙ (2/((1/x)+(1/y))) + (1/3) ∙ (√((x^2 +y^2 )/2)) ≥ xy

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y}>\mathrm{0}\:\:\mathrm{peove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{2}}{\mathrm{3}}\:\centerdot\:\frac{\mathrm{2}}{\frac{\mathrm{1}}{\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{y}}}\:+\:\frac{\mathrm{1}}{\mathrm{3}}\:\centerdot\:\sqrt{\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }{\mathrm{2}}}\:\geqslant\:\mathrm{xy} \\ $$

Question Number 151371 Answers: 1 Comments: 0

a;b∈N (a+b)^2 ∙(a-b)=128 ab=?

$$\mathrm{a};\mathrm{b}\in\mathbb{N} \\ $$$$\left(\mathrm{a}+\mathrm{b}\right)^{\mathrm{2}} \centerdot\left(\mathrm{a}-\mathrm{b}\right)=\mathrm{128} \\ $$$$\mathrm{ab}=? \\ $$

Question Number 151361 Answers: 1 Comments: 0

if x+y=45 prove that: ((1 - tan(x))/(1 - tan(y))) = tan(2y)

$$\mathrm{if}\:\:\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}=\mathrm{45}\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{1}\:-\:\mathrm{tan}\left(\mathrm{x}\right)}{\mathrm{1}\:-\:\mathrm{tan}\left(\mathrm{y}\right)}\:=\:\mathrm{tan}\left(\mathrm{2y}\right) \\ $$

Question Number 151357 Answers: 1 Comments: 6

Question Number 151351 Answers: 1 Comments: 0

Question Number 151350 Answers: 0 Comments: 4

Question Number 151346 Answers: 1 Comments: 0

x^2 + 5x + 25 = 0 ⇒ x^3 - 100 = ?

$$\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{5x}\:+\:\mathrm{25}\:=\:\mathrm{0}\:\:\Rightarrow\:\:\mathrm{x}^{\mathrm{3}} \:-\:\mathrm{100}\:=\:? \\ $$

Question Number 151344 Answers: 2 Comments: 0

x^2 - 4x + 1 = 0 ⇒ x^3 + (1/x^3 ) = ?

$$\mathrm{x}^{\mathrm{2}} \:-\:\mathrm{4x}\:+\:\mathrm{1}\:=\:\mathrm{0}\:\:\Rightarrow\:\:\mathrm{x}^{\mathrm{3}} \:+\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{3}} }\:=\:? \\ $$

Question Number 151332 Answers: 1 Comments: 2

which is larger? 2^(√(10)) or 3^2

$$\mathrm{which}\:\mathrm{is}\:\mathrm{larger}? \\ $$$$\mathrm{2}^{\sqrt{\mathrm{10}}} \:\:\mathrm{or}\:\:\mathrm{3}^{\mathrm{2}} \\ $$

Question Number 151322 Answers: 1 Comments: 1

Determine all pairs (x;y) of integers which satisfy: x^3 - 3x + y = 0

$$\mathrm{Determine}\:\mathrm{all}\:\mathrm{pairs}\:\:\left(\mathrm{x};\mathrm{y}\right)\:\:\mathrm{of}\:\mathrm{integers} \\ $$$$\mathrm{which}\:\mathrm{satisfy}: \\ $$$$\mathrm{x}^{\mathrm{3}} \:-\:\mathrm{3x}\:+\:\mathrm{y}\:=\:\mathrm{0} \\ $$

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