Question

simplify (√3-√5) (√5+√3)/7-2√5



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Answer 1

Given :–

• An equation, $\dfrac{(\sqrt{3} - \sqrt{5})(\sqrt{5} + \sqrt{3})}{7 - 2\sqrt{5}}$

Required :–

• Simplify the given equation.

Solution :–

Given equation is,

⟹ $\dfrac{(\sqrt{3} - \sqrt{5})(\sqrt{5} + \sqrt{3})}{7 - 2\sqrt{5}}$

Open both the brackets of numerator.

⟹ $\dfrac{\cancel{\sqrt{15}} + \sqrt{9} - \sqrt {25} - \cancel{\sqrt{15}}}{7 - 2 \sqrt{5} }$

⟹ $\dfrac{ \sqrt{9} - \sqrt{25}}{7 - 2 \sqrt{5} }$

We know that,

• √9 = 3
• √25 = 5

So,

⟹ $\dfrac{3 - 5}{7 - 2 \sqrt{5} }$

⟹ $\dfrac{-2}{7 - 2 \sqrt{5} }$

Now,

Multiply the additive inverse of denominator by both the numerator and the denominator.

⟹ $\dfrac{ - 2}{7 - 2 \sqrt{5} } \times \dfrac{7 + 2 \sqrt{5} }{7 + 2 \sqrt{5}}$

Again,

Using this identity.

• (a – b)(a + b) = a² – b²

⟹ $\dfrac{ - 2(7 + 2 \sqrt{5} )}{(7)^{2} + {(2\sqrt{5} )}^{2}}$

Again, square root cancel the power.

Open all the brackets.

⟹ $\dfrac{ - 14 - 4 \sqrt{5} }{49 -4 \times 5}$

⟹ $\dfrac{ - 14 - 4 \sqrt{5} }{49 - 20}$

Now, subtract the denominator,

⟹ $\dfrac{ - 14 - 4 \sqrt{5} }{29}$

Hence,

The answer of the given equation is $\dfrac{ - 14 - 4 \sqrt{5} }{29}$