Name: Introduction to Algebraic Identities
Uploaded: 2019-07-19
Duration: 4 min 19 s
Description: Polynomials can be a bit tricky to deal with at times. But if you are good with algebraic identities, they become a lot easier. So let's learn some important...

Question

If

x^{4} + \frac{1}{x^{4}} = 47

, find the value of

x^{3} + \frac{1}{x^{3}}

.

Accepted Answer

We have, $x^{4} + \frac{1}{x^{4}} = 47$

Again add and subtract $2$ on L.H.S, we get,

$\Rightarrow x^{4} + \frac{1}{x^{4}} + 2 - 2 = 47$

$\Rightarrow x^{4} + \frac{1}{x^{4}} + 2 (x^{2} \times \frac{1}{x^{2}}) = 47 + 2$

$\Rightarrow {(x^{2} + \frac{1}{x^{2}})}^{2} = 49$

$\Rightarrow x^{2} + \frac{1}{x^{2}} = 7$

Again add and subtract $2$ on L.H.S, we get

$x^{2} + \frac{1}{x^{2}} + 2 - 2 = 7$

$\Rightarrow x^{2} + \frac{1}{x^{2}} + 2 (x \times \frac{1}{x}) = 7 + 2$

$\Rightarrow {(x + \frac{1}{x})}^{2} = 9 \Rightarrow x + \frac{1}{x} = 3$

Cubing both sides, we get

$x^{3} + \frac{1}{x^{3}} + 3 \times x \times \frac{1}{x} (x + \frac{1}{x}) = 27$

$\Rightarrow x^{3} + \frac{1}{x^{3}} + 3 \times 3 = 27 \Rightarrow x^{3} + \frac{1}{x^{3}} = 18$

Thus, the value of $x^{3} + \frac{1}{x^{3}}$ is $18$ .

If $x^{4} + \frac{1}{x^{4}} = 47$ , find the value of $x^{3} + \frac{1}{x^{3}}$ .

Important Questions on Expansions

Learn and Prepare for any exam you want !

If x4+1x4=47, find the value of x3+1x3.

Important Questions on Expansions

Learn and Prepare for any exam you want !

If $x^{4} + \frac{1}{x^{4}} = 47$ , find the value of $x^{3} + \frac{1}{x^{3}}$ .