Question

rationalize the denominator.

4+3√5 by 4-3√5​

Answer 1

Step-by-step explanation:

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Answer 2

Question :

$\bullet \: \sf \dfrac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5} }$

Given :

$\bullet \: \sf \dfrac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5} }$

To do :

• Rationalize the denominator.

Solution :

• Here, We are given with a fraction and we need to rationalize the denominator of the fraction. Firstly, We will multiply the R.F(Rationalising factor) of the denominator with the fraction. And further we will use the laws of exponents wherever needed and solve it until the denominator is a rational number.

According to the question,

$\rightarrow \sf \dfrac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5} }$

Here, The R.F(Rationalising factor) for 4 - 3√5 is 4 + 3√5, Thus, We will multiply it with the fraction,

$\rightarrow \sf \dfrac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5} } \times \sf \dfrac{4 + 3 \sqrt{5} }{4 + 3 \sqrt{5} }$

Multiplying the numerator of first fraction with the second fraction and multiplying the denominator of first fraction with the second fraction,

$\rightarrow \sf \dfrac{(4 + 3 \sqrt{5} ) ^{2} }{ {(4)}^{2} - {( 3 \sqrt{5}) }^{2} }$

The numerator of the fraction seem to be like (a + b)², Thus, Let's apply the formula, a² + b² + 2ab = (a + b)², A = 4 & b = 3√5,

$\rightarrow \sf \dfrac{( {4}^{2} +{ {3 \sqrt{5} }^{2} + 2(4)(3 \sqrt{5)}) } }{ {(4)}^{2} - {( 3 \sqrt{5}) }^{2} }$

$\rightarrow \sf \dfrac{( 16 +{45 +(8)(3 \sqrt{5)}) } }{ 16 - 45 }$

$\rightarrow \sf \dfrac{16 +{45 + 24 \sqrt{5} } }{ 16 - 45 }$

$\rightarrow \sf \dfrac{61{ + 24 \sqrt{5} } }{ - 29}$

Therefore, By Rationalising the denominator of 4+3√5 by 4-3√5 we get 61 + 24√5/-29.