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The product of two binomial quadratic surds is always rational.

A) True

B) False

Answer

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Surd is an irrational root of a number. A quadratic surd is an expression where the number under the square root is rational but not a square root.

Step-1

Numbers like $\sqrt 2 $,$\sqrt 3 $,$\sqrt[3]{5}$ are called surds which satisfy the conditions of being irrational, order > 1 and having positive rational numbers as base.

Surds having order 2 is called quadratic surds.

And binomial surds are those which have two terms and one of them must be surds such as $\sqrt 2 + \sqrt 3 $,$2 + 3\sqrt 5 $.

So the general form of binomial quadratic surd is, $x + y\sqrt a $.

where, x and y = non zero rational

$\sqrt a $= quadratic surd

Step-2

We are asked if the product of two binomial quadratic surds is rational or not.

Case 1- let’s take two binomial quadratic surds as $2 + 3\sqrt 5 $and$2 - 3\sqrt 5 $

Multiplying both we get,

$(2 + 3\sqrt {5)} $$(2 - 3\sqrt {5)} $

= ${2^2} - (3\sqrt {5{)^2}} $[as we know $(a + b)(a - b) = {a^2} - {b^2}$]

= $4 - 45$

= $ - 41$

-41 is a rational number.

Step-3

Case 2- Let’s take two binomial quadratic surd as $2 + 3\sqrt 5 $and $3 + 2\sqrt 6 $

Multiplying both we get,

$(2 + 3\sqrt {5)} $$(3 + 2\sqrt 6 )$

= $(2 \times 3) + (2 \times 2\sqrt {6)} + (3 \times 3\sqrt {5)} + (3\sqrt {5 \times } 2\sqrt {6)} $

= $6 + 4\sqrt {6 + } 9\sqrt 5 + 6\sqrt {30} $

This is an irrational number.

Step-4

Hence, the product of two binomial quadratic surds are not always rational.

Step-5

The statement is false. Option (B) is correct.

If the product of two binomial surds is a rational number then both the surds are called conjugate surds.

In case 1, both the surds were conjugate surds.