Question

x + 1/x= 2, then show that :x^2+1/x^2=x^3+1/x^3=x^4+1/x^4​

Answer 1

✪AnSwEr

${\tt{\red{\underline{\large{Given}}}}}$

$\tt{X+\dfrac{1}{X}}$=2

${\sf{\green{\underline{\large{To\:find}}}}}$

• $\tt{X^3+\dfrac{1}{X^3}}$

${\sf{\pink{\underline{\Large{Explanation}}}}}$

We know that

$\boxed{\boxed{\sf\red{(a+b)^2=a^2+b^2+2ab}}}$

Therefore

$\tt{(X+\dfrac{1}{X})^2=X^2+\dfrac{1}{X^2}+2\frac{1}{X}\times{X}}$

Solving

$\tt{X+\dfrac{1}{X}}$=2

Both side squaring

$\tt{(X+\dfrac{1}{X})^2}$=(2)²

$\implies\tt{(X+\dfrac{1}{X})^2=X^2+\dfrac{1}{X^2}+2\frac{1}{X}\times{X}}=4$

$\implies\tt{4=X^2+\dfrac{1}{X^2}+2\frac{1}{X}\times{X}}$

$\implies\tt{4=X^2+\dfrac{1}{X^2}+2}$

$\implies\tt{4-2=X^2+\dfrac{1}{X^2}}$

$\implies\tt{2=X^2+\dfrac{1}{X^2}}$

Now

$\tt{X^3+\dfrac{1}{X^3}}$

Formula used

$\implies\tt{X^3+\dfrac{1}{X^3)}=(X+\dfrac{1}{x})(X^2+\dfrac{1}{X^2}-\frac{1}{X}\times{X)}}$

¶utting value

$\implies\tt{X^3+\dfrac{1}{X^3}={2}(2-1)}$

$\implies\tt{X^3+\dfrac{1}{X^3}={2}(1)}$

$\implies\tt{X^3+\dfrac{1}{X^3}=2}$

Simlilary

x^4+1/x^4 =2

Answer 2

Gɪᴠᴇɴ :-

• (x + 1/x) = 2

Tᴏ SHOW :-

• (x² + 1/x²) = (x³ + 1/x³) = (x⁴ + 1/x⁴)

Fᴏʀᴍᴜʟᴀ ᴜsᴇᴅ :-

• (a + b)² = a² + b² + 2ab
• (a + b)³ = a³ + b³ + 3ab(a + b)

Sᴏʟᴜᴛɪᴏɴ :-

→ (x + 1/x) = 2

Squaring Both sides ,

→ (x + 1/x)² = 2²

using (a + b)² = a² + b² + 2ab in LHS , we get,

→ x² + 1/x² + 2 * x * (1/x) = 4

→ (x² + 1/x²) + 2 = 4

→ (x² + 1/x²) = 4 - 2

→ (x² + 1/x²) = 2 ---------- Equation ❶

_________________________

Now,

Squaring Both sides of Equation ❶ Again we get,

→ (x² + 1/x²)² = 2²

Again using (a + b)² = a² + b² + 2ab in LHS , we get,

→ x⁴ + 1/x⁴ + 2 * x² * (1/x²) = 4

→ (x⁴ + 1/x⁴) + 2 = 4

→ (x⁴ + 1/x⁴) = 4 - 2

→ (x⁴ + 1/x⁴) = 2 ---------- Equation ❷

_________________________

Now,

→ (x + 1/x) = 2 (GIVEN).

Cubing Both sides we get,

→ (x + 1/x)³ = 2³

using (a + b)³ = a³ + b³ + 3ab(a + b) in LHS , we get,

→ x³ + 1/x³ + 3 * x * (1/x) * (x + 1/x) = 8

→ (x³ + 1/x³) + 3(x + 1/x) = 8

Putting (x + 1/x) = 2 Now,

→ (x³ + 1/x³) + 3 * 2 = 8

→ (x³ + 1/x³) = 8 - 6

→ (x³ + 1/x³) = 2 . -------------- Equation ❸

________________________

From Equation ❶ , ❷ & ❸ we can conclude That :-

→ (x² + 1/x²) = (x³ + 1/x³) = (x⁴ + 1/x⁴) (Proved).