Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 198644 by cortano12 last updated on 22/Oct/23

Commented by Rasheed.Sindhi last updated on 22/Oct/23

$$ab={\left(\frac{8-a+b}{6}\right)}^{3}or{\left(\frac{8+a-b}{6}\right)}^3?$$

Commented by cortano12 last updated on 23/Oct/23

$$\mathrm{ab}={\left(\frac{8-\mathrm{a}-\mathrm{b}}{6}\right)}^3$$

Answered by Frix last updated on 22/Oct/23

$${\left(ab\right)}^{\frac{1}{3}}=\frac{8-a-b}{6}$$$$6{\left(ab\right)}^{\frac{1}{3}}=8-(a+b)$$$$3\times 2{\left(ab\right)}^{\frac{1}{3}}=2^3-(a+b)$$$$\mathrm{assuming}{a}^{\frac{1}{3}}+b^{\frac{1}{3}}=2$$$$[a^{\frac{1}{3}}+b^{\frac{1}{3}}=c\iff a+3{\left(ab\right)}^{\frac{1}{3}}\underset{=c}{(a^{\frac{1}{3}}+b^{\frac{1}{3}})}+b=c^3]$$$$3\times (a^{\frac{1}{3}}+b^{\frac{1}{3}}){\left(ab\right)}^{\frac{1}{3}}=2^3-(a+b)$$$$3a^{\frac{2}{3}}{b}^{\frac{1}{3}}+3a^{\frac{1}{3}}{b}^{\frac{2}{3}}=2^3-(a+b)$$$$a+3a^{\frac{2}{3}}{b}^{\frac{1}{3}}+3a^{\frac{1}{3}}{b}^{\frac{2}{3}}+b=2^3$$$${(a^{\frac{1}{3}}+b^{\frac{1}{3}})}^3=2^3$$$$$$$$a^{\frac{1}{3}}+b^{\frac{1}{3}}=2$$$$a^{\frac{1}{3}}-b^{\frac{1}{3}}=10$$$$$$$$a^{\frac{1}{3}}=6\wedge b^{\frac{1}{3}}=-4$$$$a=216\wedge b=-64$$$$a-b=280$$

Commented by Frix last updated on 22/Oct/23

$$\mathrm{I}\mathrm{used}{(-r)}^{\frac{1}{3}}=-\left(r^{\frac{1}{3}}\right)$$

Commented by Frix last updated on 23/Oct/23

$$a^{\frac{1}{3}}-b^{\frac{1}{3}}={216}^{\frac{1}{3}}-{(-64)}^{\frac{1}{3}}={216}^{\frac{1}{3}}+{64}^{\frac{1}{3}}=6+4=10$$$$===$$$$ab=216\times (-64)=-13824$$$${\left(\frac{8-a-b}{6}\right)}^3={\left(\frac{8-216+64}{6}\right)}^3={(-24)}^3=-13824$$

Answered by Frix last updated on 23/Oct/23

$$\mathrm{Let}a=x^3\wedge b=y^3$$$$\left(1\right)\Rightarrow y=x-10$$$$\left(2\right)\Rightarrow$$$$x^9-45x^8+1125x^7-18360x^6+213300x^5-1798200x^4+10914048x^3-45450720x^2+114307200x-128024064=0$$$$x=6$$$$x^8-39x^7+891x^6-13014x^5+135216x^4-986904x^3+4992624x^2-15494976x+21337344=0$$$$(x^2-6x+84)(x^2-9x+21)(x^2-12x+84)(x^2-12x+144)=0$$$$x=3\pm 5\sqrt{3}\mathrm{i}\vee x=\frac{9}{2}\pm \frac{\sqrt{3}}{2}\mathrm{i}\vee x=6\pm 4\sqrt{3}\mathrm{i}\vee x=6\pm 6\sqrt{3}\mathrm{i}$$$$\mathrm{none}\mathrm{of}\mathrm{these}\mathrm{fit}\mathrm{in}\mathrm{the}\mathrm{first}\mathrm{equation}\mathrm{because}$$$$a=x^3\nLeftrightarrow a^{\frac{1}{3}}=x$$$$x=r{\mathrm{e}}^{\mathrm{i}\alpha }\Rightarrow x^3=r^3{\mathrm{e}}^{\mathrm{i}3\alpha }\mathrm{but}3\alpha =\beta \wedge -\pi <\beta ⩽\pi$$$$\arg \left({\mathrm{e}}^{\mathrm{i}3\alpha }\right)=\frac{\pi }{2}(1+\mathrm{sign}\left(\sin 3\alpha \right))+\mathrm{mod}(-3\alpha ,\pi )$$$$\Rightarrow \mathrm{i}.\mathrm{e}.$$$$x=3+5\sqrt{3}\mathrm{i}\approx 9.{16515\mathrm{e}}^{1.23732\mathrm{i}}$$$$a=x^3\approx 769.{873\mathrm{e}}^{-2.57122\mathrm{i}}$$$$a^{\frac{1}{3}}\approx 9.{16515\mathrm{e}}^{-.857072\mathrm{i}}\ne x$$$$\Rightarrow$$$$\mathrm{no}\mathrm{solution}\mathrm{with}\mathrm{rules}\mathrm{for}a,b\in \mathbb{C}$$$$1\mathrm{solution}\mathrm{with}\mathrm{rules}\mathrm{for}a,b\in \mathbb{R}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com