Question

if a =√3-√2/√3+√2
and b=√3+√2/√3-√2
find the value of a^2+b^3-5ab​

Answer 1

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❚ ANsWeR ❚

✺ Given :

• $\longrightarrow a= \dfrac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}$
• $\longrightarrow b= \dfrac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$

✺ To Find :

• a³+b³-5ab =?

✺ Explanation

(1)

$\implies ab=\dfrac{\cancel{(\sqrt{3}-\sqrt{2})}}{\cancel{(\sqrt{3}+\sqrt{2})}}\times\dfrac{\cancel{(\sqrt{3}+\sqrt{2})}}{\cancel{(\sqrt{3}-\sqrt{2})}}$

$\implies ab=1$

(2)

$\implies (a+b)=\dfrac{(\sqrt{3}-\sqrt{2})}{(\sqrt{3}+\sqrt{2})}+\dfrac{(\sqrt{3}+\sqrt{2})}{(\sqrt{3}-\sqrt{2})}$

$\implies (a+b)=\dfrac{(\sqrt{3}-\sqrt{2})^2+(\sqrt{3}+\sqrt{2})^2}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}$

$\implies(a+b)= \dfrac{2(\sqrt{3}^2+\sqrt{2}^2)}{\sqrt{3}^2-\sqrt{2}^2}$

$\implies(a+b)= \dfrac{2(3+2)}{3-2}$

$\implies (a+b)=\dfrac{10}{1}$

$\implies \boxed{(a+b)=10}$

Now , .........

✏ a³+b³-5ab

✏ (a+b)(a²-ab+b²)-5ab

✏ (a+b)(a²+b²-ab)-5ab

✏ (a+b)[(a+b)²-2ab-ab]-5ab

✏ (a+b)[(a+b)²-3ab]-5ab

✏ (10)[(10)²-3×1]-5×1

✏ (10)[100-3]-5

✏ 10×97-5

✏ 970-5

✏ 965

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✺ Therefore :

Answer 2

⠀⠀⠀⠀☛...Given...☚⠀⠀

• a = $\dfrac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} }$

• b = $\dfrac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} }$

⠀⠀ ⠀☛...To find...☚⠀ ⠀

✰✰⠀The value if a³ + b³ - 5ab

⠀ ⠀ ☛...Solution...☚⠀⠀

⠀⠀⠀In case 1st...

• a + b = $\dfrac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} } + \dfrac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} }$

• a + b = $\dfrac{( \sqrt{3} - \sqrt{2} ) ^{2} + ( \sqrt{3} + \sqrt{2} )^{2} }{( \sqrt{3} + \sqrt{2}){( \sqrt{3} - \sqrt{2} } }$

• a + b = $\dfrac{2( \sqrt{3} ^{2} + \sqrt{2} ^{2} ) ^{2} }{ \sqrt{3} ^{2} - \sqrt{2} ^{2} }$

• a + b = $\dfrac{2(3+2)}{3-2}$

• a + b = $\dfrac{10}{1}$

• a + b = 10

⠀⠀⠀In case 2nd...

• a × b = $\dfrac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} } \times \dfrac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} }$

• a × b = 1

⠀⠀⠀⠀Hence...

We have to find the value of a³ + b³ - 5ab

Substuting values in the following equation...

⠀ ⠀⠀ ⠀⠀⠀⠀ ⠀

⠀⠀ ⠀⠀⠀a³ + b³ = ( a + b) ( a² - ab + b² )

• ✂ ( a + b ) [( a + b )² - 2ab - ab ] - 5ab
• ✂ ( a + b ) [( a + b )² - 3ab ] - 5ab
• ✂ ( 10 ) [( 10 )² - 3( 1 ) ] - 5 (1)
• ✂ 10 × ( 100 - 3 ) - 5
• ✂ 10 × 97 - 5
• ✂ 970 - 5
• ✂ 965...

Therefore, the value of a³ + b³ - 5ab will be 965...