Question

If a^2+1/a^2=11, find the value of a^3+1/a^3

Answer 1

Refer to attachment:

GIVEN:

★$a^{2}+ \dfrac{1}{a^{2}} =11$

TO FIND:

★$a^{3}+\dfrac{1}{a^{3}}$

ANSWER:

WE HAVE,

=>$a^{2}+ \dfrac{1}{a^{2}} =11$

=>$a^{2}+\dfrac{1}{a^{2}} -2×a×\dfrac{1}{a}+2×a×\dfrac{1}{a}=11$

=>$(a) ^{2} + (\dfrac{1}{a^{2}}) -2×a×\dfrac{1}{a}+2=11$

=>$(a+\dfrac{1}{a})^{2}=11+2$

$\large\green{\boxed{(a+b) ^{2}=a^{2}+b^{2}+2ab}}$

=>$(a+\dfrac{1}{a})^{2}=\sqrt{13}^2$

.°.$a+\dfrac{1}{a}=\sqrt{13}$ .........,..... (1)

______________________________________

On cubing equation 1.

=>$(a+\dfrac{1}{a})^{3}=(\sqrt{13})^{3}$

$\large\purple{\boxed{(a+b) ^{3}=a^{3}+b^{3}+3ab(a+b) }}$

=>$a^{3}+\dfrac{1}{a^{3}}+3×a×\dfrac{1}{a}(a+\dfrac{1}{a})=13\sqrt{13}$

=>$a^{3}+\dfrac{1}{a^{3}}+3\sqrt{13}=13\sqrt{13}$

=>$a^{3}+\dfrac{1}{a^{3}}=13\sqrt{13}-3\sqrt{13}$

.°.$a^{3}+\dfrac{1}{a^{3}}=10\sqrt{13}$

$\huge\orange{\boxed{a^{3}+\dfrac{1}{a^{3}}=10\sqrt{13}}}$