Question

if the positive square root of (√190 +√ 80) i multiplied by (√2-1) and the
product is raised to the power of four the result would be​

Answer 1

Given :- if the positive square root of (√90 +√ 80) is multiplied by (√2-1) and the product is raised to the power of four the result would be ?

Solution :-

→ √90 + √80

→ √(9 * 10) + √(16 * 5)

→ √(3² * 10) + (4² * 5)

→ 3√10 + 4√5

→ (3*√2*√5) + (2*√2*√2*√5)

taking common ,

→ (√2 * √5)[3 + 2√2]

→ √10(3 + 2√2)

→ √10(2 + 1 + 2√2)

→ √10[(√2)² + (1)² + 2 * 1 * √2)]

comparing with (a² + b² + 2ab) = (a + b)²

→ √10[(√2 + 1)²]

now, we have given that, Positive square root of (√90 + √80) is multiply by (√2 - 1).

So,

→ +√[√10{(√2 + 1)²}] * (√2 - 1)

→ (10)^(1/4)(√2 + 1) * (√2 - 1)

using (a + b)(a - b) = a² - b²,

→ 10^(1/4) * { (√2)² - (1)²}

→ 10^(1/4) * (2 - 1)

→ 10^(1/4).

Now, given that, product is raised to the power of four.

Therefore,

→ {10^(1/4)}⁴

using (a^b)^c = (a)^( b * c) ,

→ (10)^(1/4 * 4)

→ (10)^(1)

→ 10 (Ans.)

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Answer 2

CORRECT QUESTION :

If the positive square root of (√90 +√ 80) is multiplied by (√2-1) and the product is raised to the power of four the result would be

EVALUATION

We have to determine

$\displaystyle\sf{} { \bigg[ \sqrt{ ( \sqrt{90} + \sqrt{80})} \: ( \sqrt{2} - 1) \bigg] }^{ 4 }$

Let

$\displaystyle\sf{} {P = \bigg[ \sqrt{( \sqrt{90} + \sqrt{80})} \: ( \sqrt{2} - 1) \bigg] }$

$\implies \displaystyle\sf{} {{P}^{2} = \bigg[ ( \sqrt{90} + \sqrt{80}) \: {( \sqrt{2} - 1)}^{2} \bigg] }$

$\implies \displaystyle\sf{} {{P}^{2} = \bigg[ ( \sqrt{9 \times 10} + \sqrt{8 \times 10}) \: {(2 + 1 - 2 \sqrt{2} )} \bigg] }$

$\implies \displaystyle\sf{} {{P}^{2} = \bigg[ \sqrt{10} \: ( 3 + 2 \sqrt{2} ) \: {(3- 2 \sqrt{2} )} \bigg] }$

$\implies \displaystyle\sf{} {{P}^{2} = \sqrt{10} \bigg[ \: {(3)}^{2} - {(2 \sqrt{2})}^{2} \bigg] }$

$\implies \displaystyle\sf{} {{P}^{2} = \sqrt{10} \bigg[ 9 - 8 \bigg] }$

$\implies \displaystyle\sf{} {P}^{2} = \sqrt{10}$

So

$\displaystyle\sf{} { \bigg[ \sqrt{ ( \sqrt{90} + \sqrt{80})} \: ( \sqrt{2} - 1) \bigg] }^{ 4 }$

$= \displaystyle\sf{} {P}^{4}$

$= \displaystyle\sf{} {( \sqrt{10}) }^{2}$

$= \displaystyle\sf{}10$

FINAL RESULT

$\boxed{ \displaystyle\sf{} \: \: \: { \bigg[ \sqrt{ ( \sqrt{90} + \sqrt{80})} \: ( \sqrt{2} - 1) \bigg] }^{ 4 } = 10 \: \: }$

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