If alpha and beta are two roots of the equati | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given the quadratic equation $x^{2}-64x+256=0$.From the properties of roots,$\alpha+\beta = 64$ and $\alpha\beta = 256$.We need to evaluate the ex

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(D) Given the quadratic equation $x^{2}-64x+256=0$.From the properties of roots,$\alpha+\beta = 64$ and $\alpha\beta = 256$.We need to evaluate the expression $E = \left(\frac{\alpha^{3}}{\beta^{5}}\right)^{\frac{1}{8}}+\left(\frac{\beta^{3}}{\alpha^{5}}\right)^{\frac{1}{8}}$.$E = \frac{\alpha^{3/8}}{\beta^{5/8}} + \frac{\beta^{3/8}}{\alpha^{5/8}}$.Taking the common denominator,$E = \frac{\alpha^{3/8} \cdot \alpha^{5/8} + \beta^{3/8} \cdot \beta^{5/8}}{(\alpha\beta)^{5/8}}$.$E = \frac{\alpha^{(3/8+5/8)} + \beta^{(3/8+5/8)}}{(\alpha\beta)^{5/8}} = \frac{\alpha+\beta}{(\alpha\beta)^{5/8}}$.Since $\alpha\beta = 256 = 2^{8}$,then $(\alpha\beta)^{5/8} = (2^{8})^{5/8} = 2^{5} = 32$.Substituting the values,$E = \frac{64}{32} = 2$.

If $\alpha$ and $\beta$ are two roots of the equation $x^{2}-64x+256=0$,then the value of $\left(\frac{\alpha^{3}}{\beta^{5}}\right)^{\frac{1}{8}}+\left(\frac{\beta^{3}}{\alpha^{5}}\right)^{\frac{1}{8}}$ is

Similar Questions

If the roots of the equation $x^2 + bx + c = 0$ are reciprocals of each other,then...

If the harmonic mean between the roots of $(5+\sqrt{2}) x^2-b x+(8+2 \sqrt{5})=0$ is $4$,then the value of $b$ is

If $\alpha$ and $\beta$ are the roots of the equation $ax^2 + bx + c = 0$,then what is the value of $\alpha \beta^2 + \alpha^2 \beta + \alpha \beta$?

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+ax^2+bx+c=0$,then $\alpha^{-1}+\beta^{-1}+\gamma^{-1}$ is equal to

If $\alpha$ and $\beta$ are the roots of the equation $a x^2+b x+c=0$ and if $p x^2+q x+r=0$ has roots $\frac{1-\alpha}{\alpha}$ and $\frac{1-\beta}{\beta}$,then $r$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module