Prove that root 2 +root3 is irrational?

Question

Answer

Assume it's rational. Then there are integers m,n such that m/n = √2 + √3 Square both sides: m^2 / n^2 = 5 + 2√6 'Solve' for √6: √6 = (m^2 - 5n^2) / (2n^2) so if √2 + √3 is rational, then so is √6. then prove that √6 is irrational number thus if √6 is irrational number than . √2+√3 is also irrational hope u will like my technique

Answer

Answer

Samjh nhi aa raha hai yah question

Answer

Proving that √2 * √3 is irrational:

To prove that √2 * √3 is irrational, we can use a proof by contradiction. We assume that √2 * √3 is rational and then show that it leads to a contradiction.

Assumption:

Let's assume that √2 * √3 is rational.

Expressing the assumption as an equation:

Since we assume that √2 * √3 is rational, we can express it as a fraction in the form p/q, where p and q are integers with no common factors and q is not equal to zero.

√2 * √3 = p/q

Manipulating the equation:

We can square both sides of the equation to eliminate the square roots:

(√2 * √3)^2 = (p/q)^2

2 * 3 = (p^2)/(q^2)

6 = (p^2)/(q^2)

Multiplying both sides by q^2, we get:

6q^2 = p^2

Conclusion:

From the equation 6q^2 = p^2, we can see that p^2 is divisible by 6, which implies that p is also divisible by 6.

Therefore, we can express p as p = 6k, where k is an integer.

Substituting p = 6k into the equation, we get:

6q^2 = (6k)^2

6q^2 = 36k^2

Dividing both sides of the equation by 6, we obtain:

q^2 = 6k^2

Contradiction:

From the equation q^2 = 6k^2, we can see that q^2 is also divisible by 6, which implies that q is also divisible by 6.

However, we initially assumed that p and q have no common factors, which contradicts the fact that both p and q are divisible by 6.

Final Conclusion:

Since assuming √2 * √3 to be rational leads to a contradiction, we can conclude that √2 * √3 is irrational.

Similar Class 10 Doubts

Prove that root 2 is irrational?

prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT?

Prove that root 3- root 2 is irrational ?

prove that root 2 isirrational Related: Irrational Numbers (Easy Explanation) - Real Numbers, CBSE, Class 10, Mathematics

Prove that 1/root 2 is a irrational number?

Top Courses for Class 10View all

Science Class 10

Social Studies (SST) Class 10

Mathematics (Maths) Class 10

English Grammar Advanced

English Grammar Basic

Top Courses for Class 10

Top Courses for Class 10

Suggested Free Tests

Related Content

About Class 10

Class 10 Study Material

Class 10 All Subjects

NCERT Textbook Solutions

Other Classes

Important Links

Company