Question

Expand:
$(\sqrt{5}-\sqrt{2})^{5}$

Answer 1

Answer:

145√5 - 229√2

Step-by-step explanation:

Hi,

Using the binomial expansion of (a + b)ⁿ,

where a = √5, b = - √2 and n= 5, we get

(√5 - √2)⁵

= ⁵C₀(√5)⁵ + ⁵C₁(√5)⁴(- √2) + ⁵C₂(√5)³( - √2)² + ⁵C₃(√5)²(- √2)³

+ ⁵C₄(√5)(- √2)⁴ + ⁵C₅(- √2)⁵

On simplifying the terms , we get

(√5 - √2)⁵  = 25√5 - 125√2  + 100√5 - 100√2 + 20√5 - 4√2

On simplifying we get

= 145√5 - 229√2

Hope, it helps !

Answer 2

Solution :

$(\sqrt{5}-\sqrt{2})^{5}$

=$^{5}C_{0}\left(\sqrt5 \right)^{5}-^{5}C_{1}\left(\sqrt5 \right)^{4}\left(\sqrt2 \right)\\\\+^{5}C_{2}\left(\sqrt5 \right)^{3}\left(\sqrt2\right )^{2}\\\\-^{5}C_{3}\left (\sqrt5 \right )^{2}\left(\sqrt2\right)^{3}+^{5}C_{4}\left(\sqrt5 \right )^{1}\left(\sqrt2 \right)^{5}+^{5}C_{5}\left (\sqrt2 \right)^{5}$
=$1\cdot 25\sqrt5-5\cdot 25\sqrt2+10\cdot 5\sqrt5 \cdot 2\\\\ - 10 \cdot 5 \cdot 2\sqrt2 + 5 \cdot \sqrt5 \cdot 4 - 1 \cdot 4\sqrt2$

= $25\sqrt5 - 125\sqrt2 + 100\sqrt5 - 100\sqrt2 + 20\sqrt5 - 4\sqrt2$

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