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AlgebraQuestion and Answers: Page 2

Question Number 221359 Answers: 2 Comments: 0

Question Number 221322 Answers: 3 Comments: 0

Solve for x x^(1/a) +(√x^((1/a)+(1/b)) )=x^(1/b)

$$\mathrm{Solve}\:\mathrm{for}\:{x} \\ $$$${x}^{\frac{\mathrm{1}}{{a}}} +\sqrt{{x}^{\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}} }={x}^{\frac{\mathrm{1}}{{b}}} \\ $$

Question Number 221321 Answers: 0 Comments: 0

if 0<x<y<e^2 then y^(√x) +x^2 +6xy+18y^2 +(8/x)+((16)/(9x^2 y^2 ))>2+x^(√y)

$${if}\:\mathrm{0}<{x}<{y}<{e}^{\mathrm{2}} \:{then} \\ $$$${y}^{\sqrt{{x}}} +{x}^{\mathrm{2}} +\mathrm{6}{xy}+\mathrm{18}{y}^{\mathrm{2}} +\frac{\mathrm{8}}{{x}}+\frac{\mathrm{16}}{\mathrm{9}{x}^{\mathrm{2}} {y}^{\mathrm{2}} }>\mathrm{2}+{x}^{\sqrt{{y}}} \\ $$

Question Number 221262 Answers: 2 Comments: 1

if a, b, c, > 0 , show that; (a^5 /b^2 ) + (b/c) + (c^3 /a^2 ) > 2a

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{if}\:{a},\:{b},\:{c},\:>\:\mathrm{0}\:\:,\:\mathrm{show}\:\mathrm{that}; \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{a}^{\mathrm{5}} }{{b}^{\mathrm{2}} }\:+\:\frac{{b}}{{c}}\:+\:\frac{{c}^{\mathrm{3}} }{{a}^{\mathrm{2}} }\:>\:\mathrm{2}{a} \\ $$$$ \\ $$

Question Number 221245 Answers: 1 Comments: 0

Which is greater ((√7)+1) or ((√3)+(√5))?????

$${Which}\:{is}\:{greater}\:\:\:\:\:\:\:\left(\sqrt{\mathrm{7}}+\mathrm{1}\right)\:\:{or}\:\:\left(\sqrt{\mathrm{3}}+\sqrt{\mathrm{5}}\right)????? \\ $$

Question Number 221239 Answers: 1 Comments: 0

2^x =4^y =8^z and ((1/(2x))+1(1/(4y))+(1/(6z)))=((24)/7) z=??

$$\mathrm{2}^{{x}} =\mathrm{4}^{{y}} =\mathrm{8}^{{z}} \:\:{and}\:\left(\frac{\mathrm{1}}{\mathrm{2}{x}}+\mathrm{1}\frac{\mathrm{1}}{\mathrm{4}{y}}+\frac{\mathrm{1}}{\mathrm{6}{z}}\right)=\frac{\mathrm{24}}{\mathrm{7}}\:\:\: \\ $$$${z}=?? \\ $$

Question Number 221238 Answers: 3 Comments: 0

a=5+2(√6) then {(√a)−(1/( (√a)))}=??

$${a}=\mathrm{5}+\mathrm{2}\sqrt{\mathrm{6}}\:{then}\:\left\{\sqrt{{a}}−\frac{\mathrm{1}}{\:\sqrt{{a}}}\right\}=?? \\ $$

Question Number 221195 Answers: 1 Comments: 1

Question Number 221185 Answers: 5 Comments: 0

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Question Number 221168 Answers: 1 Comments: 0

Question Number 221151 Answers: 0 Comments: 1

((√(x^2 −x−(√(x^2 −x−(√(x^2 −x−(√(...))))))))/( ((x^2 (√(x ((x^2 (√(x...))))^(1/3) ))))^(1/3) )) = (3/4) ⇒ (2/x) =?

$$\:\frac{\sqrt{{x}^{\mathrm{2}} −{x}−\sqrt{{x}^{\mathrm{2}} −{x}−\sqrt{{x}^{\mathrm{2}} −{x}−\sqrt{...}}}}}{\:\sqrt[{\mathrm{3}}]{{x}^{\mathrm{2}} \:\sqrt{{x}\:\sqrt[{\mathrm{3}}]{{x}^{\mathrm{2}} \:\sqrt{{x}...}}}}}\:=\:\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$\:\Rightarrow\:\frac{\mathrm{2}}{{x}}\:=?\: \\ $$

Question Number 221135 Answers: 1 Comments: 2

South Korean Grade 12 math Prove log_a M^n =nlog_a M Using below: When M=a^x , log_a M=x When N=a^y , log_a N=y MN=a^x ×a^y =a^(x+y) So, log_a (MN)=log_a (a^(x+y) )=x+y=log_a M+log_a N

$$\mathrm{South}\:\mathrm{Korean}\:\mathrm{Grade}\:\mathrm{12}\:\mathrm{math} \\ $$$$\mathrm{Prove}\:\mathrm{log}_{{a}} {M}^{{n}} ={n}\mathrm{log}_{{a}} {M} \\ $$$$\mathrm{Using}\:\mathrm{below}: \\ $$$$\mathrm{When}\:{M}={a}^{{x}} ,\:\mathrm{log}_{{a}} {M}={x} \\ $$$$\mathrm{When}\:{N}={a}^{{y}} ,\:\mathrm{log}_{{a}} {N}={y} \\ $$$${MN}={a}^{{x}} ×{a}^{{y}} ={a}^{{x}+{y}} \\ $$$$\mathrm{So},\:\mathrm{log}_{{a}} \left({MN}\right)=\mathrm{log}_{{a}} \left({a}^{{x}+{y}} \right)={x}+{y}=\mathrm{log}_{{a}} {M}+\mathrm{log}_{{a}} {N} \\ $$

Question Number 221017 Answers: 1 Comments: 0

Question Number 220995 Answers: 2 Comments: 0

Question Number 220987 Answers: 1 Comments: 0

Let a,b,c be positive reals such that abc=1.prove that (1/(a^3 (b+c)))+(1/(b^3 (c+a)))+(1/(c^3 (a+b)))≥(3/2)

$${Let}\:{a},{b},{c}\:{be}\:{positive}\:{reals}\:{such}\:{that}\:{abc}=\mathrm{1}.{prove}\:{that} \\ $$$$\frac{\mathrm{1}}{{a}^{\mathrm{3}} \left({b}+{c}\right)}+\frac{\mathrm{1}}{{b}^{\mathrm{3}} \left({c}+{a}\right)}+\frac{\mathrm{1}}{{c}^{\mathrm{3}} \left({a}+{b}\right)}\geqslant\frac{\mathrm{3}}{\mathrm{2}} \\ $$

Question Number 220958 Answers: 1 Comments: 1

for x, y, z >0 find the maximum of x^m y^n z^k subject to ax+by+cz=d.

$${for}\:{x},\:{y},\:{z}\:>\mathrm{0}\:{find}\:{the}\:{maximum}\:{of} \\ $$$${x}^{{m}} {y}^{{n}} {z}^{{k}} \:{subject}\:{to}\:{ax}+{by}+{cz}={d}. \\ $$

Question Number 220947 Answers: 1 Comments: 0

Σ_(k=1) ^(13) (1/(sin ((π/4)+(((k−1)π)/6))sin ((π/4)+((kπ)/6))))

$$\underset{{k}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}\:\:\frac{\mathrm{1}}{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}+\frac{\left({k}−\mathrm{1}\right)\pi}{\mathrm{6}}\right)\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}+\frac{{k}\pi}{\mathrm{6}}\right)} \\ $$

Question Number 220858 Answers: 1 Comments: 2

Question Number 220854 Answers: 2 Comments: 0

Solve for x and y 3^x +3^y =4, 3^(−x) +3^(−y ) =(4/3)

$${Solve}\:{for}\:{x}\:\:\:{and}\:\:\:\:{y} \\ $$$$\mathrm{3}^{{x}} +\mathrm{3}^{{y}} =\mathrm{4},\:\:\mathrm{3}^{−{x}} +\mathrm{3}^{−{y}\:} =\frac{\mathrm{4}}{\mathrm{3}} \\ $$

Question Number 220853 Answers: 1 Comments: 0

The two solutions of the equation are the same a(b−c)x^(2 ) +b(c−a)x+c(a−b)=0 Prove that (1/a)+(1/c)=(2/b)

$${The}\:{two}\:{solutions}\:{of}\:{the}\:{equation}\:{are}\:{the}\:{same} \\ $$$${a}\left({b}−{c}\right){x}^{\mathrm{2}\:} +{b}\left({c}−{a}\right){x}+{c}\left({a}−{b}\right)=\mathrm{0} \\ $$$${Prove}\:{that}\:\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{c}}=\frac{\mathrm{2}}{{b}} \\ $$

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Question Number 220700 Answers: 1 Comments: 0

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