Question

simplify : ​

$\frac{7 + \sqrt{8}}{7 - \sqrt{8}} + \frac{7 - \sqrt{8}}{7 + \sqrt{8}}$

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---> Step by step solution needed✔️ ​​

Answer 1

Step-by-step explanation:

We have,

$\rightarrow \bold{\dfrac{7+\sqrt{8} }{7-\sqrt{8} }\times\dfrac{7-\sqrt{8} }{7+\sqrt{8} } }$

Rationalizing $\bold{\dfrac{7+\sqrt{8} }{7-\sqrt{8} }}$ :

$\rightarrow \bold{\dfrac{7+\sqrt{8} }{7-\sqrt{8} }\times\dfrac{7+\sqrt{8} }{7+\sqrt{8} } }$

$\rightarrow \bold{\dfrac{(7+\sqrt{8})^2 }{(7)^2-(\sqrt{8})^2 } }$

Since :

• (a + b)² = a² + b² + 2ab
• (a + b)(a - b) = a² - b²

$\rightarrow \bold{\dfrac{7^2+(\sqrt{8})^2+2(7)(\sqrt{8}) }{7^2-\sqrt{8}^2 } }$

$\rightarrow \bold{\dfrac{49+8+14\sqrt{8} }{49-8} }$

Since :

• √8 × √8
• √64
• 8

$\rightarrow \bold{\dfrac{57+14\sqrt{8} }{41} }$  - (1)

Rationalizing $\bold{\dfrac{7-\sqrt{8} }{7+\sqrt{8} }}$ :

$\rightarrow \bold{\dfrac{7-\sqrt{8} }{7+\sqrt{8} } \times\dfrac{7-\sqrt{8} }{7-\sqrt{8} } }$

Similarly from above we get :

$\rightarrow \bold{\dfrac{(7-\sqrt{8})^2 }{(7+\sqrt{8}) (7-\sqrt{8}) } }$

$\rightarrow \bold{\dfrac{57-14\sqrt{8} }{41} }$ - (2)

Same denominator so add (1) and (2) :

$\rightarrow \bold{\dfrac{57+14\sqrt{8} }{41} +\dfrac{57-14\sqrt{8} }{41} }$

Taking L.C.M we get :

$\rightarrow \bold{\dfrac{57+14+57-14\sqrt{8} } {41} }$

$\rightarrow \bold{\dfrac{114+14\sqrt{8}-14\sqrt{8} }{41} }$

$\rightarrow \bold{\dfrac{114}{41} }$

∴ 114/41 is the simplified value.

Answer 2

Answer:

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