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AlgebraQuestion and Answers: Page 237

Question Number 116854 Answers: 4 Comments: 0

... calculus elementary algebra ... please solve :: ((6x+9))^(1/3) +((7−7x))^(1/3) +((x−8))^(1/3) =2 ...m.n.july.1970...

$$\:\:\:...\:\:\:{calculus}\:\:\:{elementary}\:\:{algebra}\:...\:\: \\ $$$$ \\ $$$$ \\ $$$$\:{please}\:{solve}\::: \\ $$$$ \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{6}{x}+\mathrm{9}}\:+\sqrt[{\mathrm{3}}]{\mathrm{7}−\mathrm{7}{x}}\:+\sqrt[{\mathrm{3}}]{{x}−\mathrm{8}}\:=\mathrm{2} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:...{m}.{n}.{july}.\mathrm{1970}... \\ $$$$\: \\ $$

Question Number 116824 Answers: 2 Comments: 0

Given a>b>0 , a&b real number such that a^2 −ab+b^2 =7 and a−ab+b=−1. find the value of a^2 −b^2

$$\mathrm{Given}\:\mathrm{a}>\mathrm{b}>\mathrm{0}\:,\:\mathrm{a\&b}\:\mathrm{real}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{a}^{\mathrm{2}} −\mathrm{ab}+\mathrm{b}^{\mathrm{2}} =\mathrm{7}\:\mathrm{and}\:\mathrm{a}−\mathrm{ab}+\mathrm{b}=−\mathrm{1}. \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{a}^{\mathrm{2}} −\mathrm{b}^{\mathrm{2}} \\ $$

Question Number 116780 Answers: 2 Comments: 1

Question Number 116683 Answers: 2 Comments: 0

Given 1+3+5+7=16 we know that 16=4^2 and 4 is the half of 8 which is the successor of 7. conjecture the result of this sum: 1+3+5+7+...+25

$$\mathrm{Given}\:\mathrm{1}+\mathrm{3}+\mathrm{5}+\mathrm{7}=\mathrm{16}\:\:\:\mathrm{we}\:\mathrm{know}\:\mathrm{that} \\ $$$$\mathrm{16}=\mathrm{4}^{\mathrm{2}} \:\:\mathrm{and}\:\mathrm{4}\:\mathrm{is}\:\mathrm{the}\:\mathrm{half}\:\mathrm{of}\:\mathrm{8}\:\mathrm{which}\:\mathrm{is}\:\mathrm{the}\: \\ $$$$\mathrm{successor}\:\mathrm{of}\:\mathrm{7}. \\ $$$$ \\ $$$$\mathrm{conjecture}\:\mathrm{the}\:\mathrm{result}\:\mathrm{of}\:\mathrm{this}\:\mathrm{sum}: \\ $$$$\mathrm{1}+\mathrm{3}+\mathrm{5}+\mathrm{7}+...+\mathrm{25} \\ $$

Question Number 116759 Answers: 2 Comments: 0

Question Number 116722 Answers: 1 Comments: 0

find the range of values of k for which the equation e^x −5=k has no solution

$${find}\:{the}\:{range}\:{of}\:{values}\:{of}\:{k}\:{for} \\ $$$${which}\:{the}\:{equation}\:{e}^{{x}} −\mathrm{5}={k}\:{has}\:{no} \\ $$$${solution} \\ $$

Question Number 116633 Answers: 1 Comments: 0

Question Number 116529 Answers: 1 Comments: 1

Question Number 116509 Answers: 2 Comments: 0

x^4 −48x^2 +x+565=0 x=?

$$\mathrm{x}^{\mathrm{4}} −\mathrm{48x}^{\mathrm{2}} +\mathrm{x}+\mathrm{565}=\mathrm{0}\: \\ $$$$\mathrm{x}=? \\ $$

Question Number 116507 Answers: 1 Comments: 0

Question Number 116452 Answers: 2 Comments: 0

Question Number 116358 Answers: 2 Comments: 0

Given that ((17−((27)/4)(√6)))^(1/(3 )) and ((17+((27)/4)(√6)))^(1/(3 )) are the roots of the equation x^2 −ax+b = 0. Find the value of ab.

$$\mathrm{Given}\:\mathrm{that}\:\sqrt[{\mathrm{3}\:}]{\mathrm{17}−\frac{\mathrm{27}}{\mathrm{4}}\sqrt{\mathrm{6}}}\:\mathrm{and}\:\sqrt[{\mathrm{3}\:}]{\mathrm{17}+\frac{\mathrm{27}}{\mathrm{4}}\sqrt{\mathrm{6}}} \\ $$$$\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\mathrm{x}^{\mathrm{2}} −\mathrm{ax}+\mathrm{b}\:=\:\mathrm{0}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{ab}. \\ $$

Question Number 116323 Answers: 2 Comments: 0

(1)Let a,b and c real number such that ((ab)/(a+b)) = (1/3), ((bc)/(b+c)) = (1/4) and ((ac)/(a+c)) = (1/5). Find the value of ((24abc)/(ab+ac+bc)) ? (2) Let p and q be two real number that satisfy p.q=2013. What is the minimum value of (p+q)^2 ?

$$\left(\mathrm{1}\right)\mathrm{Let}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}\:\mathrm{real}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\frac{\mathrm{ab}}{\mathrm{a}+\mathrm{b}}\:=\:\frac{\mathrm{1}}{\mathrm{3}},\:\frac{\mathrm{bc}}{\mathrm{b}+\mathrm{c}}\:=\:\frac{\mathrm{1}}{\mathrm{4}}\:\mathrm{and}\:\frac{\mathrm{ac}}{\mathrm{a}+\mathrm{c}}\:=\:\frac{\mathrm{1}}{\mathrm{5}}.\:\mathrm{Find} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{24abc}}{\mathrm{ab}+\mathrm{ac}+\mathrm{bc}}\:? \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Let}\:\mathrm{p}\:\mathrm{and}\:\mathrm{q}\:\mathrm{be}\:\mathrm{two}\:\mathrm{real}\:\mathrm{number}\:\mathrm{that} \\ $$$$\mathrm{satisfy}\:\mathrm{p}.\mathrm{q}=\mathrm{2013}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{minimum} \\ $$$$\mathrm{value}\:\mathrm{of}\:\left(\mathrm{p}+\mathrm{q}\right)^{\mathrm{2}} \:? \\ $$

Question Number 116301 Answers: 2 Comments: 0

What is the condition for a given line to 1) intersect a curve 2) be a tangent to a curve 3) not to intersect a curve

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{condition}\:\mathrm{for}\:\mathrm{a} \\ $$$$\mathrm{given}\:\mathrm{line}\:\mathrm{to}\:\: \\ $$$$\left.\mathrm{1}\right)\:\mathrm{intersect}\:\mathrm{a}\:\mathrm{curve} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{be}\:\mathrm{a}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{a}\:\mathrm{curve} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{not}\:\mathrm{to}\:\mathrm{intersect}\:\mathrm{a}\:\mathrm{curve}\: \\ $$

Question Number 116259 Answers: 1 Comments: 0

Question Number 116253 Answers: 0 Comments: 0

(1) Show that Σ_(i = 0) ^n L_i (x) = 1 (2) Show that Σ_(i = 0) ^n L_i (x). x_i ^k = x^k , k ≤ n

$$\left(\mathrm{1}\right)\:\:\:\:\mathrm{Show}\:\mathrm{that}\:\:\:\:\underset{\mathrm{i}\:\:=\:\:\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\:\mathrm{L}_{\mathrm{i}} \left(\mathrm{x}\right)\:\:\:=\:\:\:\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\mathrm{Show}\:\mathrm{that}\:\:\:\:\underset{\mathrm{i}\:\:=\:\:\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\:\mathrm{L}_{\mathrm{i}} \left(\mathrm{x}\right).\:\mathrm{x}_{\mathrm{i}} ^{\mathrm{k}} \:\:\:=\:\:\:\mathrm{x}^{\mathrm{k}} ,\:\:\:\:\:\:\:\:\mathrm{k}\:\leqslant\:\mathrm{n} \\ $$

Question Number 116221 Answers: 3 Comments: 0

Given α,β and ϕ are the roots of x^3 −px^2 +qx−pq = 0 . Find the value of (α/β)+(β/α)+(β/ϕ)+(ϕ/β)+(α/ϕ)+(ϕ/α)=?

$$\mathrm{Given}\:\alpha,\beta\:\mathrm{and}\:\varphi\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$$\mathrm{x}^{\mathrm{3}} −\mathrm{px}^{\mathrm{2}} +\mathrm{qx}−\mathrm{pq}\:=\:\mathrm{0}\:. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{\alpha}{\beta}+\frac{\beta}{\alpha}+\frac{\beta}{\varphi}+\frac{\varphi}{\beta}+\frac{\alpha}{\varphi}+\frac{\varphi}{\alpha}=? \\ $$

Question Number 116192 Answers: 2 Comments: 1

Question Number 116173 Answers: 0 Comments: 2

How many ways can the letters of the word DENKENMATHEMATICAL be arranged if no same letters must be together

$$\mathrm{How}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{of}\:\mathrm{the}\:\mathrm{word}\:\:\mathrm{DENKENMATHEMATICAL} \\ $$$$\mathrm{be}\:\mathrm{arranged}\:\mathrm{if}\:\mathrm{no}\:\mathrm{same}\:\mathrm{letters}\:\mathrm{must}\:\mathrm{be}\:\mathrm{together} \\ $$

Question Number 116163 Answers: 1 Comments: 0

I need a general rule for factorising 1) a^n +b^n 2) a^n −b^n when n is even and when n is odd.

$$\mathrm{I}\:\mathrm{need}\:\mathrm{a}\:\mathrm{general}\:\mathrm{rule}\:\mathrm{for}\:\mathrm{factorising} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{a}^{\mathrm{n}} +\mathrm{b}^{\mathrm{n}} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{a}^{\mathrm{n}} −\mathrm{b}^{\mathrm{n}} \\ $$$$\mathrm{when}\:\mathrm{n}\:\mathrm{is}\:\mathrm{even}\:\mathrm{and}\:\mathrm{when}\:\mathrm{n}\:\mathrm{is}\:\mathrm{odd}. \\ $$

Question Number 116138 Answers: 3 Comments: 1

If α and β are the roots of the quadratic equation x^2 −10x+2=0 and α >β, find: (i) (1/β)−(1/α) (ii)α^3 −β^3

$$\mathrm{If}\:\:\alpha\:\mathrm{and}\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$$\mathrm{the}\:\mathrm{quadratic}\:\mathrm{equation}\: \\ $$$$\mathrm{x}^{\mathrm{2}} −\mathrm{10x}+\mathrm{2}=\mathrm{0}\:\mathrm{and}\:\alpha\:>\beta,\:\mathrm{find}: \\ $$$$\left(\mathrm{i}\right)\:\frac{\mathrm{1}}{\beta}−\frac{\mathrm{1}}{\alpha} \\ $$$$\left(\mathrm{ii}\right)\alpha^{\mathrm{3}} −\beta^{\mathrm{3}} \\ $$

Question Number 116226 Answers: 2 Comments: 1

(1/(2+(√2))) +(1/(3(√2)+2(√3))) +(1/(4(√3)+3(√4)))+...+(1/(100(√(99))+99(√(100))))

$$\frac{\mathrm{1}}{\mathrm{2}+\sqrt{\mathrm{2}}}\:+\frac{\mathrm{1}}{\mathrm{3}\sqrt{\mathrm{2}}+\mathrm{2}\sqrt{\mathrm{3}}}\:+\frac{\mathrm{1}}{\mathrm{4}\sqrt{\mathrm{3}}+\mathrm{3}\sqrt{\mathrm{4}}}+...+\frac{\mathrm{1}}{\mathrm{100}\sqrt{\mathrm{99}}+\mathrm{99}\sqrt{\mathrm{100}}} \\ $$

Question Number 116092 Answers: 1 Comments: 2

Solve for x and y if x^y =36

$${Solve}\:{for}\:{x}\:{and}\:{y}\:{if} \\ $$$${x}^{{y}} =\mathrm{36} \\ $$

Question Number 116091 Answers: 2 Comments: 0

Is there a formular to tell how many times a digit occur in an interval. e.g. How many times digits 2 occur between 1 − 100

$$\mathrm{Is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{formular}\:\mathrm{to}\:\mathrm{tell}\:\mathrm{how}\:\mathrm{many}\:\mathrm{times}\:\mathrm{a}\:\mathrm{digit}\:\mathrm{occur}\:\mathrm{in}\:\mathrm{an}\:\mathrm{interval}. \\ $$$$ \\ $$$$\mathrm{e}.\mathrm{g}.\:\:\mathrm{How}\:\mathrm{many}\:\mathrm{times}\:\mathrm{digits}\:\:\mathrm{2}\:\:\mathrm{occur}\:\mathrm{between}\:\:\mathrm{1}\:−\:\mathrm{100} \\ $$

Question Number 116059 Answers: 2 Comments: 0

Σ_(n=1) ^∞ ((n!)/3^(n+1) )

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}!}{\mathrm{3}^{{n}+\mathrm{1}} } \\ $$

Question Number 116057 Answers: 2 Comments: 0

Σ_(n=2) ^∞ (3/(3n+1))=?

$$\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\mathrm{3}}{\mathrm{3}{n}+\mathrm{1}}=? \\ $$

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