Question

please give me the solution ​

Answer 1

Here we go hon,

${a}^{3} + \frac{1}{ {a}^{3} } = 18$

Now,

${(a + \frac{1}{a}) }^{3} = {a}^{3} + \frac{1}{ {a}^{3} } + 3(a)( \frac{1}{a} )(a + \frac{1}{a} ) \\ {(a + \frac{1}{a}) }^{3} = 18 + 3(a + \frac{1}{a} )$

Let a + (1/a) = x for the sake of making the equation look less scarier,

${x}^{3} = 18 + 3x \\ {x}^{3} - 3x = 8 \\ x( {x}^{2} - 3) = 18$

Now you're supposed to solve a cubic equation here but I'm gonna go for hit and trial.

18 can be written as
1×18
2×9
3×6
6×3
9×2
18×1

The third value satisfies x as 3,

So,
x must be equal to 3.

Now,
$a + \frac{1}{a} = 3$

Squaring both sides gives,

${a}^{2} + \frac{1}{ {a}^{2} } + 2 = 9 \\ {a}^{2} + \frac{1}{ {a}^{2} } = 7$

Once again squaring both sides gives,

${{a}^{4} + \frac{1}{ {a}^{4} } }^{} + 2 = 49 \\ \\ {a}^{4} + \frac{1}{ {a}^{4} } = 47$

Hence, your answer is 47.