if a² +1/ a² = 23 find the value of a³ + 1/a³ ( hint: b² = ac)
Answers
EXPLANATION.
⇒ a² + 1/a² = 23.
As we know that,
⇒ (a + 1/a)² = a² + 1/a² + 2(a)(1/a).
⇒ (a + 1/a)² = a² + 1/a² + 2.
Put the value of a² + 1/a² = 23 in the equation, we get.
⇒ (a + 1/a)² = 23 + 2.
⇒ (a + 1/a)² = 25.
⇒ (a + 1/a) = √25.
⇒ (a + 1/a) = 5.
As we know that,
⇒ (a³ + 1/a³) = (a + 1/a)(a² + 1/a² - (a)(1/a)).
⇒ (a³ + 1/a³) = (a + 1/a)(a² + 1/a² - 1).
Put the values in the equation, we get.
⇒ (a³ + 1/a³) = (5)(23 - 1).
⇒ (a³ + 1/a³) = 5(22).
⇒ (a³ + 1/a³) = 110.
Answer:
110
Step-by-step explanation:
Given,
a²+ 1/a² = 23
Applying the formula: (a + b)² = a² + b² + 2ab
a² + b² = (a + b)² - 2ab
Putting the given value:
23 = (a + 1/a )² - ( 2 × a × 1/a )
23 = (a + 1/a )² - 2
23 + 2 = (a + 1/a )²
(a + 1/a ) = √25
(a + 1/a ) = 5
To find: a³ + 1/a³
Applying the formula: (a + b)³ = a³ + b³ + 3a²b + 3ab²
(a + 1/a )³ = a³ + 1/a³ + ( 3 × a² × 1/a ) + ( 3 × a × 1/a² )
(5)³ = a³ + 1/a³ + 3a + 3/a
125 = a³ + 1/a³ + 3 (a + 1/a )
125 = a³ + 1/a³ + 3 (5)
125 = a³ + 1/a³ + 15
a³ + 1/a³ = 125 - 15
a³ + 1/a³ = 110