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Chapter 3: Problem 218

If \(\alpha, \beta\) are roots of \(8 \mathrm{x}^{2}-3 \mathrm{x}+27=0\) then \(\left(\alpha^{2} / \beta\right)^{(1 / 3)}+\left(\beta^{2} / \alpha\right)^{(1 / 3)}=\) (a) \((1 / 3)\) (b) \((7 / 2)\) (c) 4 (d) \((1 / 4)\)

Short Answer

Expert verified
The sum \(\left(\alpha^{2} / \beta\right)^{1/3} + \left(\beta^{2} / \alpha\right)^{1/3}\) is an imaginary number, so none of the given options (a), (b), (c) or (d) is the correct answer for this exercise.

Step by step solution

01

Identify the Quadratic Equation and Roots

Firstly, we are given the quadratic equation \(8x^{2}-3x+27=0\) with roots α and β.
02

Use Vieta's Formulas for the Sum and Product of Roots

We know that for a quadratic equation \(ax^{2}+bx+c=0\), the sum of the roots is given by \(-b/a\) and the product of the roots by \(c/a\). So for our equation, the sum and product of the roots are: $$\alpha + \beta = \frac{-(-3)}{8} = \frac{3}{8}$$ $$\alpha \beta = \frac{27}{8}$$
03

Multiply the given expression by its conjugate

We want to find the value of \(\left(\alpha^{2} / \beta\right)^{1/3} + \left(\beta^{2} / \alpha\right)^{1/3}\). To make the expression easier to work with, let's multiply the expression by the conjugate of the given expression: $$\left(\left(\alpha^{2} / \beta\right)^{1/3} + \left(\beta^{2} / \alpha\right)^{1/3}\right)\left(\left(\alpha^{2} / \beta\right)^{1/3} - \left(\beta^{2} / \alpha\right)^{1/3}\right)$$
04

Simplify the expression

Now let's simplify the expression we got in step 3: $$\left(\frac{\alpha^{2}}{\beta}\right) - \left(\frac{\beta^{2}}{\alpha}\right) = \frac{\alpha^{3}\beta - \alpha\beta^{3}}{\alpha\beta} = \frac{(\alpha^{3} - \beta^{3})}{\alpha\beta}$$
05

Use the sum and product of roots

Now, let's substitute the relationship we found in step 2 into the expression we got in step 4: \[\frac{(\alpha+\beta)(\alpha^2-\alpha\beta+\beta^2)}{\alpha\beta} =\frac{\frac{3}{8}\left[(\alpha+\beta)^2-3\alpha\beta\right]}{\frac{27}{8}} = \frac{3[(\frac{3}{8})^2 - \frac{9}{8}\frac{27}{8}]}{27}\]
06

Calculate the desired value

Now let's calculate the value of the expression above: \[\frac{3\left[(\frac{9}{64}) - \frac{243}{64}\right]}{27} = \frac{3(\frac{-234}{64})}{27} = \frac{-234}{64} \cdot \frac{1}{9} = -\frac{26}{8} = \frac{-13}{4}\]
07

Find the value of the initial expression

Since we multiplied our initial expression by its conjugate, we need to find the square root of the result we got in step 6 to get the answer for the initial expression. As the result is negative, the sum \(\left(\alpha^{2} / \beta\right)^{1/3} + \left(\beta^{2} / \alpha\right)^{1/3}\) is an imaginary number, which means none of the given options (a), (b), (c) or (d) is the correct answer for this exercise.

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