Quadratic Equations
Symmetric Functions of Roots
Grade 11
Question:
<p>Given that the quadratic equation \(x^2 - 64x + 256 = 0\) has roots \(a\) and \(b\), find the value of \(\left(\frac{a^3}{b^5}\right)^{1/8} + \left(\frac{b^3}{a^5}\right)^{1/8}\).</p>
Step-by-Step Solution
Key Concept: Use Vieta's formulas to find sum and product of roots, then simplify the given expression using exponent properties.
<p><strong>Solution:</strong></p><p>From the equation $x^2 - 64x + 256 = 0$:</p><p>$a + b = 64$ and $ab = 256$</p><p>$$\left(\frac{a^3}{b^5}\right)^{1/8} + \left(\frac{b^3}{a^5}\right)^{1/8} = \frac{a + b}{(ab)^{5/8}} = \frac{64}{(256)^{5/8}} = \frac{64}{2^5} = \frac{64}{32} = 2$$</p>
Correct Answer: 2