Question

Thiis is a Question
$03. Find \: a \: and \: b, If \frac{7 - 4 \sqrt{3} }{7 + 4 \sqrt{3} } =a+b√3$
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Answer 1

Answer

• The value of a = 97 and b = -56.

Given

• $\sf\cfrac{7 - 4 \sqrt{3} }{7 + 4 \sqrt{3} } =a+b \sqrt{3}$

To Find

• The value of a and b.

Step By Step Explanation

• Step 1.
• To rationalize the denominator.

First we need to rationalize the denominator. So let's do it !!

$\longmapsto \sf\cfrac{7 - 4 \sqrt{3} }{7 + 4 \sqrt{3} } =a+b \sqrt{3} \\ \\ \longmapsto \sf\cfrac{7 - 4 \sqrt{3} }{7 + 4 \sqrt{3} } \: \: \times \: \: \cfrac{7 - 4 \sqrt{3} }{7 - 4 \sqrt{3} } =a+b \sqrt{3} \\ \\ \longmapsto \sf\cfrac{({7 - 4 \sqrt{3})}^{2} }{ {(7)}^{2} - {(4 \sqrt{3})}^{2} } =a+b \sqrt{3} \\ \\ \longmapsto \sf\cfrac{49 + 48 - 56 \sqrt{3} }{49 - 48} = a + b \sqrt{3} \\ \\ \longmapsto \sf 97 - 56 \sqrt{3} = a + b \sqrt{3}$

• Step 2.
• To find the value of a and b.

Now we need to find the value of a and b. So let's do it !!

$\longmapsto\sf97 - 56 \sqrt{3} = a + b \sqrt{3}$

From here we conclude that ↓

$\longmapsto \sf \underline{ \boxed{ \bold{ \green{97 = a}}}} \: \: \: and \: \: \: \underline{\boxed{\bold{ \pink{ - 56 = b}}}} \: \: \: \: \: \: \bigstar$

Therefore, the value of a = 97 and b = -56.

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