Question

Rationalize the Denominator

4/7+ 4 √3

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Answer 1

Answer:

28 - 16√3

Step-by-step explanation:

Rationalize the denominator means making the denominator a rational number. In order to rationalize the denominator, we multiply the given fraction with the rationalizing factor of the denominator with both the numerator and the denominator of the given fraction.

Here, we have to rationalize the denominator of :

$\longmapsto \rm { \dfrac{4}{7 + 4\sqrt{3} } }$

The denominator of the given fraction is in the form of (a + b). Rationalizing factor of a term means changing the sign to its opposite sign, Rationalizing factor of (a + b) is (a - b). So, the rationalizing factor of (7 + 4√3) is (7 - 4√3). Multiplying (7 - 4√3) with both the numerator and the denominator.

$\longmapsto \rm { \dfrac{4}{7 + 4\sqrt{3} } \times \dfrac{7 - 4 \sqrt{3} }{7 - 4 \sqrt{3} } }$

Rearranging the terms.

$\longmapsto \rm { \dfrac{4(7 - 4 \sqrt{3})}{(7 + 4\sqrt{3})(7 - 4 \sqrt{3}) } }$

Multiplying 4 with the terms in the brackets in the numerator & and using the identity given below in the denominator.

• (a + b)(a - b) = a² - b²

Simplifying further in the denominator,

$\longmapsto \rm { \dfrac{28 - 16\sqrt{3}}{ {(7)}^{2} - {(4\sqrt{3})}^{2} } }$

Putting the values of squares of the numbers in the denominator.

$\longmapsto \rm { \dfrac{28 - 16\sqrt{3}}{ 49 - {48}} }$

Subtracting the numbers in the denominator,

$\longmapsto \rm { \dfrac{ 28 -16\sqrt{3} }{ 1} }$

∴ Hence, Denominator is rationalized !