Question

Simplify : $\frac{2\sqrt{6} }{\sqrt{2} + \sqrt{3} } + \frac{6\sqrt{2} }{\sqrt{6} +\sqrt{3} } - \frac{8\sqrt{3} }{\sqrt{6} +\sqrt{2} }$

Answer 1

Solution

Given:-

• 2√6/(√2+√3) + 6√2/(√6+√3) - 8√3/(√6+√2)

Find:-

• Value of given terms

Explanation

☞2√6/(√2+√3) + 6√2/(√6+√3) - 8√3/(√6+√2)

Rationalize denominators of all term

➠ 2√6(√2-√3)/(√2-√3)(√2+√3) + 6√2(√6-√3)/(√6-√3)(√6+√3) - 8√3.(√6-√2)/(√6-√2)(√6+√2)

☛ Using required Formula

★ (x-y)(x+y) = x² - y²

So,

➠ 2√6.(√2-√3)/[(√2)²-(√3)³] + 6√2(√6-√3)/[(√6)²-(√3)²] - 8√3.(√6-√2)/[(√6)²-(√2)²]

➠ 2√6(√2-√3)/(2-3) + 6√2(√6-√3)/(6-3) - 8√3.(√6-√2)/(6-2)

➠ 2√6.(√3-√2) + 2√2.(√6-√3) - 2√3.(√6-√2)

➠ 2.√18 - 2√12 + 2.√12 - 2.√6 - 2.√18 + 2.√6

➠ ( 2.√18 - 2.√18) + (2.√12 - 2.√12) + (2.√16 - 2.√16)

➠ 0 + 0 + 0

➠ 0 [ Ans.]

________________

Answer 2

$\underline{\underline\textbf{Question...}}$

⠀⠀⠀⠀ ⠀ ⠀⠀ ⠀⠀⠀

✮ Simplify...

⠀⠀⠀⠀ ⠀ ⠀⠀ ⠀⠀

$\frac{2\sqrt{6} }{\sqrt{2} + \sqrt{3} } + \frac{6\sqrt{2} }{\sqrt{6} +\sqrt{3} } - \frac{8\sqrt{3} }{\sqrt{6} +\sqrt{2} }$

⠀⠀⠀⠀ ⠀ ⠀⠀ ⠀⠀⠀

$\underline{\underline\textbf{Solution...}}$

⠀⠀⠀⠀ ⠀ ⠀⠀ ⠀⠀⠀

Rationalize the denominator...

⠀⠀⠀⠀ ⠀ ⠀⠀ ⠀⠀⠀

$\frac{2\sqrt{6} }{\sqrt{2} + \sqrt{3} } + \frac{6\sqrt{2} }{\sqrt{6} +\sqrt{3} } - \frac{8\sqrt{3} }{\sqrt{6} +\sqrt{2} }$

⠀⠀⠀⠀ ⠀ ⠀⠀ ⠀⠀

${\overbrace{\underbracebrace{(a^{2} - b^{2})\:=\:(a - b)(a + b)}}}$

⠀⠀⠀⠀ ⠀ ⠀⠀ ⠀⠀⠀

$\frac{2 \sqrt{6} }{ \sqrt{2} + \sqrt{3} } \times \frac{1}{ \sqrt{2} - \sqrt{3} } + \frac{6 \sqrt{2} }{ \sqrt{6} + \sqrt{3} } \times \frac{1}{ \sqrt{6} - \sqrt{3} } - \frac{8 \sqrt{3} }{ \sqrt{6} + \sqrt{2} } \times \frac{1}{ \sqrt{6} - \sqrt{2} }$

⠀⠀⠀⠀ ⠀ ⠀⠀ ⠀⠀⠀

---➣ $\frac{2 \sqrt{6} }{( \sqrt{2}) ^{2} - {( \sqrt{3} })^{2} } + \frac{6 \sqrt{2} }{ {( \sqrt{6}) }^{2} - { (\sqrt{3} })^{2} } - \frac{8 \sqrt{3} }{ { (\sqrt{6} )}^{2} - { (\sqrt{2} })^{2} }$

⠀⠀⠀⠀ ⠀ ⠀⠀ ⠀⠀⠀

---➣ 2√6 × (√3-√2) + 2√2 × (√6-√3) - 2√3 × (√6-√2)

⠀⠀⠀⠀ ⠀ ⠀⠀ ⠀⠀⠀

---➣ 2 × √18 - 2√12 + 2 × √12 - 2.√6 - 2 × √18 + 2 × √6

⠀⠀⠀⠀ ⠀ ⠀⠀ ⠀⠀⠀

---➣ ( 2 × √18 - 2 ×√18) + (2 × √12 - 2 × √12) + (2 × √16 - 2 × √16)

⠀⠀⠀⠀ ⠀ ⠀⠀ ⠀⠀⠀

---➣ 0 + 0 + 0

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---➣ 0