Please I want solution and answer

1.	Please I want solution and answer
Answer» Question : - $\sf{ \sqrt[4]{81} - 8 \sqrt[3]{216} + 15 \sqrt[5]{32} + \sqrt{225} }$ ANSWER Given : - $\sf{ \sqrt[4]{81} - 8 \sqrt[3]{216} + 15 \sqrt[5]{32} + \sqrt{225} }$ Required to find : - Value of $\sf{ \sqrt[4]{81} - 8 \sqrt[3]{216} + 15 \sqrt[5]{32} + \sqrt{225} }$ Solution : - $\sf \to \sqrt[4]{81} - 8 \sqrt[3]{216} + 15 \sqrt[5]{32} + \sqrt{225} \\ \\ \rm we \: know \: that \\ \\ \to \sf \sqrt[a]{x} = {x}^{ \frac{1}{a} } \\ \\ \to \sf (81 {)}^{ \frac{1}{4} } - 8(216 {)}^{ \frac{1}{ 3} } + 15(32 {)}^{ \frac{1}{5} } + (225 {)}^{ \frac{1}{2} } \\ \\ \to \sf ( {3}^{4} {)}^{ \frac{1}{4} } - 8( {6}^{3} {)}^{ \frac{1}{ 3} } + 15( {2}^{5} {)}^{ \frac{1}{5} } + ( {15}^{2} {)}^{ \frac{1}{2} } \\ \\ \rm \because ( {a}^{m} {)}^{n} = {a}^{mn} \\ \\ \to \sf {3}^{ \frac{4}{4} } - 8( {6}^{ \frac{3}{3} } ) + 15( {2}^{ \frac{5}{5} } ) + {15}^{ \frac{2}{2} } \\ \\ \to \to \sf 3 - 8(6) + 15(2) + 15 \\ \\ \to 3 - 48 + 30 + 15 \\ \\ \to 48 - 48 \\ \\ \to 0$ $\rule{400}{4}$ Question : - If x = 3 + √8 , then the value of ( x² + 1/x² ) Answer Given : - x = 3 + √8 Required to find : - value of ( x² + 1/x² ) ? Solution : - x = 3 + √8 ➞ x² = ➞ ( 3 + √8 )² Using the identity ; ( a + b )² = a² + b² + 2ab ➞ ( 3 + √8 )² = ➞ ( 3 )² + ( √8 )² + 2 ( 3 ) ( √8 ) ➞ 9 + 8 + 6√8 ➞ 17 + 6√8 1/x² = ➞ 1/ 17 + 6√8 Here we NEED to rationalize the denominator . ➞ Rationalising factor = 17 - 6√8 MULTIPLY the numerator and denominator with 17 - 6√8 ➞ 1/17 + 6√8 x 17 - 6√8/17 - 6√8 ➞ 17 - 6√8/( 17 )² - ( 6√8 )² ➞ 17 - 6√8/289 - 288 ➞ 17 - 6√8/1 ➞ 17 - 6√8 value of x² + 1/x² is ; ➞ 17 + 6√8 + 17 - 6√18 ➞ 17 + 17 ➞ 34 $\rule{400}{4}$ Question : - A rational NUMBER between √2 and √3 Answer Given : - √2 & √3 Required to find : - A rational number in between them ? Formula used : - A rational number between any 2 rational numbers a & b is a+b/2 SOLUTION : - A rational number √2 and √3 is ; Using the formula , A rational number between any 2 rational numbers a & b is a+b/2 ➞ a + b/2 ➞ (√2 + √3)/2 Hence, ➞ The rational number between √2 and √3 is (√2 +√3)/2

Answer»

Question : -

$\sf{ \sqrt[4]{81} - 8 \sqrt[3]{216} + 15 \sqrt[5]{32} + \sqrt{225} }$

ANSWER

Given : -

$\sf{ \sqrt[4]{81} - 8 \sqrt[3]{216} + 15 \sqrt[5]{32} + \sqrt{225} }$

Required to find : -

Value of

$\sf{ \sqrt[4]{81} - 8 \sqrt[3]{216} + 15 \sqrt[5]{32} + \sqrt{225} }$

Solution : -

$\sf \to \sqrt[4]{81} - 8 \sqrt[3]{216} + 15 \sqrt[5]{32} + \sqrt{225} \\ \\ \rm we \: know \: that \\ \\ \to \sf \sqrt[a]{x} = {x}^{ \frac{1}{a} } \\ \\ \to \sf (81 {)}^{ \frac{1}{4} } - 8(216 {)}^{ \frac{1}{ 3} } + 15(32 {)}^{ \frac{1}{5} } + (225 {)}^{ \frac{1}{2} } \\ \\ \to \sf ( {3}^{4} {)}^{ \frac{1}{4} } - 8( {6}^{3} {)}^{ \frac{1}{ 3} } + 15( {2}^{5} {)}^{ \frac{1}{5} } + ( {15}^{2} {)}^{ \frac{1}{2} } \\ \\ \rm \because ( {a}^{m} {)}^{n} = {a}^{mn} \\ \\ \to \sf {3}^{ \frac{4}{4} } - 8( {6}^{ \frac{3}{3} } ) + 15( {2}^{ \frac{5}{5} } ) + {15}^{ \frac{2}{2} } \\ \\ \to \to \sf 3 - 8(6) + 15(2) + 15 \\ \\ \to 3 - 48 + 30 + 15 \\ \\ \to 48 - 48 \\ \\ \to 0$

$\rule{400}{4}$

Question : -

If x = 3 + √8 , then the value of ( x² + 1/x² )

Answer

Given : -

x = 3 + √8

Required to find : -

value of ( x² + 1/x² ) ?

Solution : -

x = 3 + √8

➞ x² =

➞ ( 3 + √8 )²

Using the identity ;

( a + b )² = a² + b² + 2ab

➞ ( 3 + √8 )² =

➞ ( 3 )² + ( √8 )² + 2 ( 3 ) ( √8 )

➞ 9 + 8 + 6√8

➞ 17 + 6√8

1/x² =

➞ 1/ 17 + 6√8

Here we NEED to rationalize the denominator .

➞ Rationalising factor = 17 - 6√8

MULTIPLY the numerator and denominator with 17 - 6√8

➞ 1/17 + 6√8 x 17 - 6√8/17 - 6√8

➞ 17 - 6√8/( 17 )² - ( 6√8 )²

➞ 17 - 6√8/289 - 288

➞ 17 - 6√8/1

➞ 17 - 6√8

value of x² + 1/x² is ;

➞ 17 + 6√8 + 17 - 6√18

➞ 17 + 17

➞ 34

$\rule{400}{4}$

Question : -

A rational NUMBER between √2 and √3

Answer

Given : -

√2 & √3

Required to find : -

A rational number in between them ?

Formula used : -

A rational number between any 2 rational numbers a & b is a+b/2

SOLUTION : -

A rational number √2 and √3 is ;

Using the formula ,

A rational number between any 2 rational numbers a & b is a+b/2

➞ a + b/2

➞ (√2 + √3)/2

Hence,

➞ The rational number between √2 and √3 is (√2 +√3)/2