Question

Pls help me with ths question ​

Answer 1

Answer:

irrational

Step-by-step explanation:

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Answer 2

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✤ Required Answer:

► GiveN:

• x = 7 + 4√3

► To FinD:

• x² - 1/x²
• And whether it is rational or irrational...?

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✤ How to solve?

First of all, we need to find the rationalised form of 1/x and then after this, we'll have x and 1/x. Now, we can find x + 1/x and x - 1/x, this will be used to find x² - 1/x² by the identity:

• (a + b)(a - b) = a² - b²

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✤ Solution:

We have,

➝ x = 7 + 4√3

Then,

➝ 1/x = 1/7 + 4√3

Rationalising,

➝ 1/x = 7 - 4√3 / (7 + 4√3)(7 - 4√3)

➝ 1/x = 7 - 4√3 / 7² - (4√3)²

➝ 1/x = 7 - 4√3 / 49 - 48

➝ 1/x = 7 - 4√3 / 1

➝ 1/x = 7 - 4√3

Now,

❒ x + 1/x = 7 + 4√3 + (7 - 4√3)

➝ x + 1/x = 7 + 4√3 + 7 - 4√3

➝ x + 1/x = 14

❒ x - 1/x = 7 + 4√3 - (7 - 4√3)

➝ x - 1/x = 7 + 4√3 - 7 + 4√3

➝ x - 1/x = 8√3

By using (a + b)(a - b) = a² - b² identity,

• a = x
• b = 1/x

➝ (x + 1/x)(x - 1/x) = x² - 1/x²

➝ 14 × 8√3 = x² - 1/x²

➝ 112√3 = x² - 1/x²

Flipping it,

➝ x² - 1/x² = 112√3

Here,

112√3 is a irrational number because it is the product of 112(rational) and √3(irrational) and we know that product of a rational and a irrational is always irrational$.$

► So, the result is a irrational number.

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