Q1.If α (alpha) and β (beta) be two roots of the equation x² – 64x + 256 = 0. Then the value of -
(a.) (α³/β⁵)¹/⁸ + (β³/α⁵)¹/⁸
(b.) (256α/α² + 256) + (256β/β² + 256)
Answers
Explanation:
✬ Values = 2 & 8 Respectively ✬
Step-by-step explanation:
Given:
Alpha and beta are the two roots of equation.
Equation is x² – 64x + 256 = 0
To Find:
Value of
(α³/β⁵)¹/⁸ + (β³/α⁵)¹/⁸
(256α/α² + 256) + (256β/β² + 256)
Solution: Basic formulae and concepts to be used here
α + β = –b/a
αβ = c/a
m⁵ × m³ = m(³ + ⁵) ← If bases are same then powers will be added.
m² × n² = (mn)² ← If bases are different and powers are same.
Let's solve the first one -
➟ (α³/β⁵)¹/⁸ + (β³/α⁵)¹/⁸
➟ α⅜ / β⅝ + β⅜ / α⅝ { taking LCM }
➟ α⅜ × β⅝ + β⅝ × β⅜ / β⅝ × α⅝
➟ α¹ + β¹ / αβ⅝
➟ α + β / αβ⅝
➟ –b/a / (c/a)⅝
➟ –(–64) / (256)⅝
➟ 64 / {(16)²}⅝
➟ 64 / [{(2)⁴}²]⅝
➟ 64 / 2⁵
➟ 64 / 32 = 2
Hence, the value of (α³/β⁵)¹/⁸ + (β³/α⁵)¹/⁸ is 2.
_________________________
Let's move to the second question.
(256α/α² + 256) + (256β/β² + 256)
➮ Given equation is x² – 64x + 256 = 0
Putting the value α in the place or x in above equation.
We got α² – 64α + 256 = 0
➮ α² – 64α + 256 = 0
➮ α² + 256 = 64α
Multiplying both sides by 4.
➮ 4 × (α² + 256) = 4 × 64α
➮ 4 x (α² + 256) = 256α
➮ 4 = 256α / α² + 256ㅤㅤㅤㅤㅤ(eqⁿ i )
Now again
Putting the value β in the place or x in above equation.
We got β² – 64β + 256 = 0
➯ β² – 64β + 256 = 0
➯ β² + 256 = 64β
Multiplying both sides by 4.
➯ 4 × (β² + 256) = 4 × 64β
➯ 4 = 256β / β² + 256ㅤㅤㅤㅤㅤ(eqⁿ ii )
Let's put the values of both equations in the second question.
⟹ (256α/α² + 256) + (256β/β² + 256)
⟹ 4 + 4
=8
Hence, the value of (256α/α² + 256) + (256β/β² + 256) is 8.
