Question

if(a+1/a)=3thena³+1/a³ equals to​

Answer 1

Given :  $\sf{ a + {\dfrac{1}{a}} = 3 }$

To find :  $\sf{a^3 + {\dfrac{1}{a^3}} }$

Solution :  Cubing both sides of $\sf{ a + {\dfrac{1}{a}} = 3. }$

$\longrightarrow\sf{ \left( a + {\dfrac{1}{a}} \right) ^3 = (3)^3 }$

${\boxed{\sf{\bullet \ Identity \ : \ (x + y)^3 = x^3 + y^3 + 3xy(x + y) }}}$

$\sf{\qquad \quad Here, \ x = a, \ b = {\dfrac{1}{a}} }$

$\longrightarrow\sf{(a)^3 + \left( {\dfrac{1}{a}} \right) ^3 + 3.a. {\dfrac{1}{a}} \left( a + {\dfrac{1}{a}} \right) = 27}$

$\longrightarrow\sf{a^3 + {\dfrac{1}{a^3}} + 3 \left( a + {\dfrac{1}{a}} \right) = 27}$

• Putting known values.

$\longrightarrow\sf{a^3 + {\dfrac{1}{a^3}} + 3(3) = 27}$

$\longrightarrow\sf{a^3 + {\dfrac{1}{a^3}} + 9 = 27}$

$\longrightarrow\sf{a^3 + {\dfrac{1}{a^3}} = 27 - 9}$

$\longrightarrow\sf{a^3 + {\dfrac{1}{a^3}} = {\boxed{18}}}$