Question

Find the value of $a^3+\frac{1}{a^3} \hspace{3} when \hspace{3} a^2+\frac{1}{a^2}=7,a\ \textgreater \ 0$

Answer 1

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Answer 2

Answer:

18

Step-by-step explanation:

Hi,

Given a² + 1/a² = 7

Adding 2 on both sides of the equation, we get

a² + 1/a² + 2 = 7 + 2 = 9

(a)² + (1/a)² + 2 (a)(1/a) = 9

⇒(a + 1/a)² = 9

⇒ a + 1/a = ±3

Since a > 0, 1/a will be positive and a + 1/a > 0

Hence , the value of a + 1/a = 3.

Consider a³ + 1/a³

We know that x³ + y³ = ( x + y )*(x² + y² - xy)

So we can write a³ + 1/a³

= (a + 1/a)*(a² + 1/a² -1)

= 3*(7 - 1)

= 3*6

= 18

Hence, a³ + 1/a³ = 18.

Hope, it helps !