Question

examine whether(2-root2) (2+root2) are rational or irrational​

Answer 1

Answer:

Step-by-step explanation:

(2-root2) (2+root2)

(a+b) (a-b) =a²-b²

2²-(root2)²

4-2=2

That is rational number.

Answer 2

Question: Examine whether $(2-\sqrt{2} )(2+\sqrt{2} )$ is rational or irrational.

Answer:

The expression $(2-\sqrt{2} )(2+\sqrt{2} )$ is rational.

Step-by-step explanation:

Irrational number: The number which cannot be written in the p/q form, q≠0, where p and q are integers.

Rational number: The number which can be written in the p/q form, q≠0, where p and q are integers.

Recall the identity,

$(a+b)(a-b)=a^2-b^2$

Step 1 of 1

As we know that,

The number 2 is a rational number and the number $\sqrt{2}$ is an irrational number.

To check: $(2-\sqrt{2} )(2+\sqrt{2} )$ is rational or irrational.

Consider the expression as follows:

$(2-\sqrt{2} )(2+\sqrt{2} )$

Here, $a=2$ and $b=\sqrt{2}$

Simplify the expression using the identity as follows:

⇒ $2^2-(\sqrt{2} )^2$

⇒ $4-2$

⇒ $2$

⇒ $(2-\sqrt{2} )(2+\sqrt{2} )=2$

Notice that the number $2$ is a rational number.

Therefore, the expression $(2-\sqrt{2} )(2+\sqrt{2} )$ is rational.

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