Answered

Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Our platform provides a seamless experience for finding reliable answers from a knowledgeable network of professionals. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.

What are the value of a and b respectively, if (3+3√5)/(3+√5) - (7-3√5)/(3-√5) = a+√5b ?​

What Are The Value Of A And B Respectively If 33535 73535 A5b class=

Sagot :

xKelvin

Answer:

[tex]\displaystyle a = 0 \text{ and } b = 1[/tex]

Step-by-step explanation:

We are given that:

[tex]\displaystyle \frac{7 + 3\sqrt{5}}{3 + \sqrt{5}} - \frac{7 - 3\sqrt{5}}{ 3 - \sqrt{5}} = a + \sqrt{5b}[/tex]

And we want to determine the values of a and b.

Simplify the fractions. We can multiply each by the conjugate of its denominator:

[tex]\displaystyle \frac{7 + 3\sqrt{5}}{3 + \sqrt{5}} \left(\frac{3-\sqrt{5}}{3-\sqrt{5}}\right) - \frac{7 - 3\sqrt{5}}{3 - \sqrt{5}} \left( \frac{ 3 + \sqrt{5}}{3 + \sqrt{5}}\right)[/tex]

Simplify:

[tex]\displaystyle = \frac{(7 + 3\sqrt{5})(3 - \sqrt{5})}{(3)^2 - (\sqrt{5})^2} - \frac{(7 - 3\sqrt{5})(3 + \sqrt{5})}{(3)^2 - (\sqrt{5})^2} \\ \\ \\ = \frac{(21 - 7\sqrt{5} + 9\sqrt{5} - 3(5))}{(9) - (5)} - \frac{(21+ 7\sqrt{5} - 9\sqrt{5} -3(5))}{(9) - (5)} \\ \\ \\ = \frac{21 + 2\sqrt{5} - 15}{4} - \frac{21 -2\sqrt{5}-15}{4} \\ \\ \\ = \frac{(6 + 2\sqrt{5})-(6-2\sqrt{5})}{4} \\ \\ \\ = \frac{4\sqrt{5}}{4} \\ \\ = \sqrt{5} = 0 + \sqrt{5(1)}[/tex]

In conclusion, a = 0 and b = 1.

SalmonWorldTopper

[tex]\large \sf \underline{Solution - }[/tex]

[tex] \rm Given \: : \: \dfrac{7 + 3 \sqrt{5} }{3 + \sqrt{5} } - \dfrac{7 - 3 \sqrt{5} }{3 - \sqrt{5} }= a + \sqrt{5b} [/tex]

  • Here, we have to find the value of a and b.

➢ Multiplying the conjugate of their denominator to both the fractions.

So,

[tex] \rm \: \implies \dfrac{7 + 3 \sqrt{5} }{3 + \sqrt{5} } - \dfrac{7 - 3 \sqrt{5} }{3 - \sqrt{5} }= a + \sqrt{5b} [/tex]

[tex] \rm \: \implies \dfrac{(7 + 3 \sqrt{5})(3 - \sqrt{5}) }{(3 + \sqrt{5} )(3 - \sqrt{5})} - \dfrac{(7 - 3 \sqrt{5}) (3 + \sqrt{5})}{(3 - \sqrt{5})(3 + \sqrt{5}) }= a + \sqrt{5b} [/tex]

We know,

[tex] \rm\implies (a - b)(a + b) = (a)^{2} - (b)^{2}[/tex]

Then,

[tex] \rm \: \implies \dfrac{(7 + 3 \sqrt{5})(3 - \sqrt{5}) }{3^{2} - ( \sqrt{5} )^{2} } - \dfrac{(7 - 3 \sqrt{5}) (3 + \sqrt{5})}{3^{2} - ( \sqrt{5} )^{2} } = a + \sqrt{5b} [/tex]

[tex] \rm \: \implies \dfrac{(7 + 3 \sqrt{5})(3 - \sqrt{5}) }{9- 5 } - \dfrac{(7 - 3 \sqrt{5}) (3 + \sqrt{5})}{9 - 5 } = a + \sqrt{5b} [/tex]

[tex] \rm \: \implies \dfrac{(7 + 3 \sqrt{5})(3 - \sqrt{5}) }{4} - \dfrac{(7 - 3 \sqrt{5}) (3 + \sqrt{5})}{4 } = a + \sqrt{5b} [/tex]

Combine both the fractions,

[tex] \rm \: \implies \dfrac{(7 + 3 \sqrt{5})(3 - \sqrt{5}) - (7 - 3 \sqrt{5}) (3 + \sqrt{5})}{4} = a + \sqrt{5b} [/tex]

[tex] \rm \: \implies \dfrac{21-7\sqrt{5}+9\sqrt{5}-15-21-7\sqrt{5}+9\sqrt{5}+15}{4}= a + \sqrt{5b} [/tex]

[tex] \rm \: \implies \dfrac{4 \sqrt{5} }{4}= a + \sqrt{5b} [/tex]

[tex] \rm \: \implies \sqrt{5} = a + \sqrt{5b} [/tex]

[tex] \rm \: : \implies 0 + \sqrt{5 \times 1} = a + \sqrt{5b} [/tex]

Therefore, we notice that

  • The value of a is 0
  • The value of b is 1
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.