Question

(√x)^8/3×(√x)^10/3÷3(√x)^4/3×(√x)^2/3=?
ans: x^2/3
​

Answer 1

Answer:

Since 2+i

3

is a root and the coefficients are real, the second root has to be 2−i

3

Thus, (x−2−i

3

)(x−2+i

3

)=x

2

−2x+ix

3

−2x+4−i2

3

−ix

3

+i2

3

+3

=x

2

−4x+7 is a root of the given equation.

Dividing the original equation by x

2

−4x+7, we have

(x

4

−4x

2

+8x+35)/(x

2

−4x+7)=x

2

+4x+5

Thus, we obtain the other root as x

2

+4x+5

Again factorizing this root, we get x=

2

−4±

16−20

=

2

−4±2i

=−2±i

The four roots are therefore 2+i

3

,2−i

3

,−2+i,−2−i

Answer 2

${\sqrt{x}}^{\frac{8}{3}}\times \frac{{\sqrt{x}}^{\frac{10}{3}}}{3}{\sqrt{x}}^{\frac{4}{3}}{\sqrt{x}}^{\frac{2}{3}}$

${x}^{\frac{1\times 8}{2\times 3}}\times \frac{{\sqrt{x}}^{\frac{10}{3}}}{3}{\sqrt{x}}^{\frac{4}{3}}{\sqrt{x}}^{\frac{2}{3}}$

${x}^{\frac{8}{2\times 3}}\times \frac{{\sqrt{x}}^{\frac{10}{3}}}{3}{\sqrt{x}}^{\frac{4}{3}}{\sqrt{x}}^{\frac{2}{3}}$

${x}^{\frac{8}{6}}\times \frac{{\sqrt{x}}^{\frac{10}{3}}}{3}{\sqrt{x}}^{\frac{4}{3}}{\sqrt{x}}^{\frac{2}{3}}$

${x}^{\frac{4}{3}}\times \frac{{\sqrt{x}}^{\frac{10}{3}}}{3}{\sqrt{x}}^{\frac{4}{3}}{\sqrt{x}}^{\frac{2}{3}}$

${x}^{\frac{4}{3}}\times \frac{{x}^{\frac{1\times 10}{2\times 3}}}{3}{\sqrt{x}}^{\frac{4}{3}}{\sqrt{x}}^{\frac{2}{3}}$

${x}^{\frac{4}{3}}\times \frac{{x}^{\frac{ 10}{2\times 3}}}{3}{\sqrt{x}}^{\frac{4}{3}}{\sqrt{x}}^{\frac{2}{3}}$

${x}^{\frac{4}{3}}\times \frac{{x}^{\frac{ 10}{6}}}{3}{\sqrt{x}}^{\frac{4}{3}}{\sqrt{x}}^{\frac{2}{3}}$

${x}^{\frac{4}{3}}\times \frac{{x}^{\frac{5}{3}}}{3}{\sqrt{x}}^{\frac{4}{3}}{\sqrt{x}}^{\frac{2}{3}}$

${x}^{\frac{4}{3}}\times \frac{{x}^{\frac{5}{3}}}{3}{x}^{\frac{1\times 4}{2\times 3}}{\sqrt{x}}^{\frac{2}{3}}$

${x}^{\frac{4}{3}}\times \frac{{x}^{\frac{5}{3}}}{3}{x}^{\frac{ 4}{2\times 3}}{\sqrt{x}}^{\frac{2}{3}}$

${x}^{\frac{4}{3}}\times \frac{{x}^{\frac{5}{3}}}{3}{x}^{\frac{ 4}{6}}{\sqrt{x}}^{\frac{2}{3}}$

${x}^{\frac{4}{3}}\times \frac{{x}^{\frac{5}{3}}}{3}{x}^{\frac{2}{6}}{\sqrt{x}}^{\frac{2}{3}}$

${x}^{\frac{4}{3}}\times \frac{{x}^{\frac{5}{3}}}{3}{x}^{\frac{2}{3}}{\sqrt{x}}^{\frac{2}{3}}$

${x}^{\frac{4}{3}}\times \frac{{x}^{\frac{5}{3}}}{3}{x}^{\frac{2}{3}}{x}^{\frac{1\times 2}{2\times 3}}$

${x}^{\frac{4}{3}}\times \frac{{x}^{\frac{5}{3}}}{3}{x}^{\frac{2}{3}}{x}^{\frac{ 2}{2\times 3}}$

${x}^{\frac{4}{3}}\times \frac{{x}^{\frac{5}{3}}}{3}{x}^{\frac{2}{3}}{x}^{\frac{ 2}{6}}$

${x}^{\frac{4}{3}}\times \frac{{x}^{\frac{5}{3}}}{3}{x}^{\frac{2}{3}}\sqrt[3]{x}$

$\frac{{x}^{\frac{4}{3}}{x}^{\frac{5}{3}}{x}^{\frac{2}{3}}\sqrt[3]{x}}{3}$

$\frac{{x}^{\frac{4}{3}+\frac{5}{3}+\frac{2}{3}+\frac{1}{3}}}{3}$

$\frac{{x}^{4}}{3}$