Key Concepts and Formulas

• Vieta's Formulas: For a quadratic equation $ax^2 + bx + c = 0$, the sum of the roots is $\alpha + \beta = -\frac{b}{a}$ and the product of the roots is $\alpha\beta = \frac{c}{a}$.
• Exponent Rules:
  • $(a^m)^n = a^{mn}$
  • $\frac{a^m}{a^n} = a^{m-n}$
  • $(ab)^n = a^n b^n$

Step-by-Step Solution

Step 1: Find the sum and product of the roots using Vieta's formulas.

The given quadratic equation is $x^2 - 64x + 256 = 0$. Here, $a = 1$, $b = -64$, and $c = 256$.

• The sum of the roots is: $\alpha + \beta = -\frac{b}{a} = -\frac{-64}{1} = 64$
• The product of the roots is: $\alpha\beta = \frac{c}{a} = \frac{256}{1} = 256$

Explanation: Vieta's formulas provide a direct way to determine the sum and product of the roots from the coefficients of the quadratic equation, without needing to solve for the individual roots. This is a crucial first step to simplify the target expression.*

Step 2: Simplify the given expression.

We are asked to find the value of ${\left( {{{{\alpha ^3}} \over {{\beta ^5}}}} \right)^{1/8}} + {\left( {{{{\beta ^3}} \over {{\alpha ^5}}}} \right)^{1/8}}$

Applying the exponent rule $(a/b)^n = a^n/b^n$ and $(a^m)^n = a^{mn}$, we have: $= \frac{\alpha^{3/8}}{\beta^{5/8}} + \frac{\beta^{3/8}}{\alpha^{5/8}}$

Explanation: Applying exponent rules allows us to rewrite the original expression in a more manageable form. This step simplifies further manipulations.*

Step 3: Combine the terms.

To add the two fractions, we find a common denominator, which is $\alpha^{5/8} \beta^{5/8} = (\alpha\beta)^{5/8}$.

$= \frac{\alpha^{3/8} \cdot \alpha^{5/8}}{\beta^{5/8} \cdot \alpha^{5/8}} + \frac{\beta^{3/8} \cdot \beta^{5/8}}{\alpha^{5/8} \cdot \beta^{5/8}}$

Using the exponent rule $a^m \cdot a^n = a^{m+n}$, we have:

$= \frac{\alpha^{3/8 + 5/8}}{(\alpha\beta)^{5/8}} + \frac{\beta^{3/8 + 5/8}}{(\alpha\beta)^{5/8}}$ $= \frac{\alpha^{8/8}}{(\alpha\beta)^{5/8}} + \frac{\beta^{8/8}}{(\alpha\beta)^{5/8}}$ $= \frac{\alpha}{(\alpha\beta)^{5/8}} + \frac{\beta}{(\alpha\beta)^{5/8}}$ $= \frac{\alpha + \beta}{(\alpha\beta)^{5/8}}$

Explanation: Combining the fractions into a single term allows us to substitute the values obtained from Vieta's formulas in the next step.*

Step 4: Substitute the values of $\alpha + \beta$ and $\alpha\beta$.

We know that $\alpha + \beta = 64$ and $\alpha\beta = 256$. Substituting these values into the expression, we get:

$= \frac{64}{(256)^{5/8}}$

Since $256 = 2^8$, we can rewrite the denominator as:

$(256)^{5/8} = (2^8)^{5/8} = 2^{8 \cdot \frac{5}{8}} = 2^5 = 32$

Therefore,

$= \frac{64}{32} = 2$

Explanation: Substituting the values and simplifying the expression leads to the final numerical answer.*

Common Mistakes & Tips

• Careless Exponent Manipulation: Ensure you correctly apply the exponent rules, especially when dealing with fractional exponents. A common mistake is incorrectly simplifying expressions like $(a^m)^n$.
• Incorrectly Calculating $\alpha\beta$: Always double-check that you are using the correct formula for the product of roots, $\alpha\beta = c/a$.
• Arithmetic Errors: Be careful with basic arithmetic operations.

Summary

By using Vieta's formulas to find the sum and product of the roots, and then simplifying the given expression using exponent rules, we found the value of the expression to be 2.

The final answer is \boxed{2}, which corresponds to option (C).