Question

4. If a+1/a = 3, find (i) a^2 + 1/a^2 (ii) a^4 +1/a^4​

Answer 1

Answer:

1st》7 , 2nd 》47

Step-by-step explanation:

QUESTION 1:

$a + \frac{1}{a} = 3 \: \: find \: \: {a}^{2} + \frac{1}{ {a}^{2} }$

SOLUTION:

$( {a + \frac{1}{a} })^{2} = {3}^{2} \: \: \: \: \: \: \: \: ((a + {b})^{2} = {a}^{2} + 2ab + {b}^{2} ) \\ = {a}^{2} + 2a \frac{1}{a} + { \frac{1}{a} }^{2} = 9 \\ {a}^{2} + 2 + { \frac{1}{a} }^{2} = 9 \\ {a}^{2} + { \frac{1}{a} }^{2} = 9 - 2 \\ {a}^{2} + { \frac{1}{a} }^{2} = 7$

QUESTION 2

$find \: {a}^{4} + { \frac{1}{a} }^{4}$

solution :

${a}^{2} + { \frac{1}{a} }^{2} = 7 \\ ( { {a}^{2} + \frac{1}{ {a}^{2} } })^{2} \: \: \: using \: same \: identity \\ { {a}^{2} }^{2} + 2 {a}^{2} { \frac{1}{a} }^{2} + { { \frac{1}{a} }^{2} }^{2} = {7}^{2} \\ {a}^{4} + 2 + { \frac{1}{a} }^{4} = {7}^{2} \\ {a}^{4} + { \frac{1}{a} }^{4} = 49 - 2 \\ {a}^{4} + { \frac{1}{a} }^{4} = 47$