Prove that 3+2√5 is irrational Video Lecture | Mathematics (Maths) Class 10

Ans. To prove that 3√2 is irrational, we need to assume the opposite, which is that it is rational. So, let's assume that 3√2 is rational and can be expressed as a fraction p/q, where p and q are integers with no common factors other than 1. By squaring both sides of the equation, we get 18 = (p^2)/(q^2). This implies that p^2 = 18q^2. Since the left-hand side is divisible by 3, p^2 must also be divisible by 3. This means p itself must be divisible by 3. Let p = 3k, where k is an integer. Substituting this back into the equation, we get (3k)^2 = 18q^2, which simplifies to 9k^2 = 18q^2. Dividing both sides by 9, we get k^2 = 2q^2. Now, this implies that q^2 is even, which means q must also be even. But if both p and q are even, they would have a common factor of 2, which contradicts our assumption. Therefore, our assumption that 3√2 is rational is incorrect, and thus it is irrational.

Ans. An irrational number is a real number that cannot be expressed as a fraction p/q, where p and q are integers, and q is not equal to zero. Irrational numbers have decimal representations that neither terminate nor repeat. They are non-repeating and non-terminating decimals.

Ans. No, we cannot simplify 3√2 to a rational number. The square root of 2 is already an irrational number, and multiplying it by 3 does not change its irrationality. Therefore, 3√2 remains an irrational number and cannot be expressed as a fraction.

Ans. Yes, all square roots of non-perfect squares are irrational. A non-perfect square is a number whose square root is not an integer. When we take the square root of a non-perfect square, we get an irrational number. This is because if the square root could be expressed as a fraction, it would mean that the original number is a perfect square.

Ans. Yes, we can approximate the value of 3√2 using decimal approximations. However, since 3√2 is an irrational number, its decimal representation would be a non-repeating and non-terminating decimal. We can use calculators or computer programs to calculate an approximation of 3√2 to a desired number of decimal places.