Question

Factorise x^2 + 1/x^2 =98, then find the value of x^3 + 1/x^3

Answer 1

Given

$\to \sf \bigg( {x}^{2} + \dfrac{1}{ {x}^{2} } \bigg) = 98$

To Find

$\to \sf\bigg( {x}^{3} + \dfrac{1}{ {x}^{3} } \bigg)$

We Know that

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

So We can write as

$\sf \to\bigg(x + \dfrac{1}{x} \bigg) ^{2} = {x}^{2} + \dfrac{1}{ {x}^{2} } + 2 \times x \times \dfrac{1}{x}$

$\sf \to\bigg(x + \dfrac{1}{x} \bigg) ^{2} = {x}^{2} + \dfrac{1}{ {x}^{2} } + 2$

$\sf \to \bigg(x + \dfrac{1}{x} \bigg) ^{2} = 98 + 2$

$\sf \to \bigg(x + \dfrac{1}{x} \bigg) ^{2} = 100$

$\sf \to \bigg(x + \dfrac{1}{x} \bigg) ^{} = 10$

Now we have to find

$\to \sf\bigg( {x}^{3} + \dfrac{1}{ {x}^{3} } \bigg)$

Use this identities

$\sf \to(a + b)^{3} = {a}^{3} + {b}^{3} + 3ab(a + b)$

We get

$\sf \to \: \bigg(x + \dfrac{1}{x} \bigg) ^{3} = {x}^{3} + \dfrac{1}{ {x}^{3} } + 3 \times x \times \dfrac{1}{x} \bigg(x + \dfrac{1}{x} \bigg)$

$\sf \to(10) {}^{3} = {x}^{3} + \dfrac{1}{ {x}^{3} } + 3 (10)$

$\sf \to \: 1000 = {x}^{3} + \dfrac{1}{ {x}^{3} } + 30$

$\sf \to \: 1000 - 30= {x}^{3} + \dfrac{1}{ {x}^{3} }$

$\sf \to \: {x}^{3} + \dfrac{1}{ {x}^{3} } = 970$

Answer

$\sf \to \: {x}^{3} + \dfrac{1}{ {x}^{3} } = 970$

Answer 2

Answer:

x^3+1/x^3 = 970

Step-by-step explanation:

Correct Question:

If x^2 + 1/x^2 =98, then find the value of x^3 + 1/x^3.

Identities involved:

1. (a+b)^2 = a^2+b^2+2ab
2. a^3+b^3 = (a+b)(a^2+b^2-ab)

Solution:

First, we should use the first identity:

➽ (x+1/x)^2 = x^2+(1/x)^2+2x(1/x)

➽ (x+1/x)^2 = 98+2

➽ (x+1/x)^2 = 100

➽ x+1/x = √100

➽ x+1/x = 10

Now, use the second identity:

x^3+1/x^3 = (x+1/x)(x^2+1/x^2 - x(1/x))

➽ x^3+1/x^3 = (10)(98-1)

➽ x^3+1/x^3 = (10)(97)

➽ x^3+1/x^3 = 970

Conclusion:

The value of x^3+1/x^3 is 970.