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#### Question 1:

Rationalise the denominator of each of the following (i-vii):

(i) $\frac{3}{\sqrt{5}}$

(ii) $\frac{3}{2\sqrt{5}}$

(iii) $\frac{1}{\sqrt{12}}$

(iv) $\frac{\sqrt{2}}{\sqrt{5}}$

(v) $\frac{\sqrt{3}+1}{\sqrt{2}}$

(vi) $\frac{\sqrt{2}+\sqrt{5}}{3}$

(vii) $\frac{3\sqrt{2}}{\sqrt{5}}$

(i) We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(ii) We know that rationalization factor foris. We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(iii) We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(iv) We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(v) We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(vi) We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(vii) We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

#### Question 2:

Find the value to three places of decimals of each of the following. It is given that
and $\sqrt{10}=3.162$.

(i) $\frac{2}{\sqrt{3}}$

(ii) $\frac{3}{\sqrt{10}}$

(iii) $\frac{\sqrt{5}+1}{\sqrt{2}}$

(iv) $\frac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}$

(v) $\frac{2+\sqrt{3}}{3}$

(vi) $\frac{\sqrt{2}-1}{\sqrt{5}}$

(i) We know that rationalization factor of the denominator is. We will multiply numerator and denominator of the given expression by, to get

The value of expression can be round off to three decimal places as.

Hence the given expression is simplified to.

(ii) We know that rationalization factor of the denominator is . We will multiply numerator and denominator of the given expression by , to get

The value of expression can be round off to three decimal places as.

Hence the given expression is simplified to.

(iii) We know that rationalization factor of the denominator is. We will multiply numerator and denominator of the given expression by, to get

The value of expression can be round off to three decimal places as.

Hence the given expression is simplified to.

(iv) We know that rationalization factor of the denominator is. We will multiply numerator and denominator of the given expression by, to get

The value of expression can be round off to three decimal places as.

Hence the given expression is simplified to.

(v) Given that

Putting the value of, we get

The value of expression can be round off to three decimal places as.

Hence the given expression is simplified to.

(vi) We know that rationalization factor of the denominator is. We will multiply numerator and denominator of the given expression by, to get

Putting the value of and, we get

The value of expression can be round off to three decimal places as.

Hence the given expression is simplified to.

#### Question 3:

Express each one of the following with rational denominator:

(i) $\frac{1}{3+\sqrt{2}}$

(ii) $\frac{1}{\sqrt{6}-\sqrt{5}}$

(iii) $\frac{16}{\sqrt{41}-5}$

(iv) $\frac{30}{5\sqrt{3}-3\sqrt{5}}$

(v) $\frac{1}{2\sqrt{5}-\sqrt{3}}$

(vi) $\frac{\sqrt{3}+1}{2\sqrt{2}-\sqrt{3}}$

(vii) $\frac{6-4\sqrt{2}}{6+4\sqrt{2}}$

(viii) $\frac{3\sqrt{2}+1}{2\sqrt{5}-3}$

(ix) $\frac{{b}^{2}}{\sqrt{{a}^{2}+{b}^{2}}+a}$

(i) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(ii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(iii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(iv) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(v) We know that rationalization factor for is.We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(vi) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(vii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(viii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(ix) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to .

#### Question 4:

Rationales the denominator and simplify:

(i) $\frac{3-\sqrt{2}}{3+\sqrt{2}}$

(ii) $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}$

(iii) $\frac{1+\sqrt{2}}{3-2\sqrt{2}}$

(iv) $\frac{2\sqrt{6}-\sqrt{5}}{3\sqrt{5}-2\sqrt{6}}$

(v) $\frac{4\sqrt{3}+5\sqrt{2}}{\sqrt{48}+\sqrt{18}}$

(vi) $\frac{2\sqrt{3}-\sqrt{5}}{2\sqrt{2}+3\sqrt{3}}$

(i) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(ii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(iii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(iv) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(v) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(vi) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

#### Question 5:

Simplify:

(i) $\frac{5+\sqrt{3}}{5-\sqrt{3}}+\frac{5-\sqrt{3}}{5+\sqrt{3}}$

(ii) $\frac{1}{2+\sqrt{3}}+\frac{2}{\sqrt{5}-\sqrt{3}}+\frac{1}{2-\sqrt{5}}$

(iii) $\frac{2}{\sqrt{5}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{3}{\sqrt{5}+\sqrt{2}}$

(i) We know that rationalization factor forand areand respectively. We will multiply numerator and denominator of the given expression and by and respectively, to get

Hence the given expression is simplified to.

(ii) We know that rationalization factor forand areand respectively. We will multiply numerator and denominator of the given expression and by and respectively, to get

Hence the given expression is simplified to.

(iii) We know that rationalization factor forand areand respectively. We will multiply numerator and denominator of the given expression and by and respectively, to get

Hence the given expression is simplified to.

#### Question 6:

In each of the following determine rational numbers a and b:

(i) $\frac{\sqrt{3}-1}{\sqrt{3}+1}=a-b\sqrt{3}$

(ii) $\frac{4+\sqrt{2}}{2+\sqrt{2}}=n-\sqrt{b}$

(iii) $\frac{3+\sqrt{2}}{3-\sqrt{2}}=a+b\sqrt{2}$

(iv) $\frac{5+3\sqrt{3}}{7+4\sqrt{3}}=a+b\sqrt{3}$

(v) $\frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}}=a-b\sqrt{77}$

(vi)    $\frac{4+3\sqrt{5}}{4-3\sqrt{5}}=a+b\sqrt{5}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

(i) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

On equating rational and irrational terms, we get

Hence, we get.

(ii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

On equating rational and irrational terms, we get

Hence, we get.

(iii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

On equating rational and irrational terms, we get

Hence, we get

(iv) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

On equating rational and irrational terms, we get

Hence, we get.

(v) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

On equating rational and irrational terms, we get

Hence, we get.

(vi) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

On equating rational and irrational terms, we get

Hence, we get.

#### Question 7:

Find the value of $\frac{6}{\sqrt{5}-\sqrt{3}}$, it being given that $\sqrt{3}=1.732$ and $\sqrt{5}=2.236$

We know that rationalization factor for is . We will multiply denominator and numerator of the given expression by , to get

Putting the values of and, we get

Hence value of the given expression is.

#### Question 8:

Find the values of each of the following correct to three places of decimals, it being given that $\sqrt{2}=1.4142,$ $\sqrt{3}=1.732$, $\sqrt{5}=2.2360,$ $\sqrt{6}=2.4495$ and $\sqrt{10}=3.162$,

(i) $\frac{3-\sqrt{5}}{3+2\sqrt{5}}$

(ii) $\frac{1+\sqrt{2}}{3-2\sqrt{2}}$

(i) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Putting the values of, we get

Hence the given expression is simplified to.

(ii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Putting the value of, we get

Hence the given expression is simplified to.

#### Question 9:

Simplify:

(i) $\frac{3\sqrt{2}-2\sqrt{3}}{3\sqrt{2}+2\sqrt{3}}+\frac{\sqrt{12}}{\sqrt{3}-\sqrt{2}}$
(ii) $\frac{7+3\sqrt{5}}{3+\sqrt{5}}-\frac{7-3\sqrt{5}}{3-\sqrt{5}}$

(i) We know that rationalization factor forand areand respectively. We will multiply numerator and denominator of the given expression and by and respectively, to get

Hence the given expression is simplified to.

(ii) We know that rationalization factor forand areand respectively. We will multiply numerator and denominator of the given expression and by and respectively, to get

Hence the given expression is simplified to.

#### Question 10:

If x = 2+$\sqrt{3}$, find the value of ${x}^{3}+\frac{1}{{x}^{3}}$

We know that. We have to find the value of.

As therefore,

We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Putting the value of and , we get

Hence the value of the given expression

#### Question 11:

If x = 3+$\sqrt{8}$, find the value of ${x}^{2}+\frac{1}{{x}^{2}}$

We know that. We have to find the value of . As therefore,

We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Putting the value of x and , we get

Hence the given expression is simplified to.

#### Question 12:

If find the value of $4{x}^{3}+2{x}^{2}-8x+7$.

We have,

It can be simplified as

On squaring both sides, we get

The given equation can be rewritten as.

Therefore, we have

Hence, the value of given expression is.

#### Question 1:

$\sqrt{10}×\sqrt{15}$ is equal to

(a) 5$\sqrt{6}$

(b) 6$\sqrt{5}$

(c) $\sqrt{30}$

(d) $\sqrt{25}$

Given that, it can be simplified as

Therefore given expression is simplified and correct choice is

#### Question 2:

is equal to

(a) $\sqrt[5]{36}$

(b) $\sqrt[5]{6×0}$

(c) $\sqrt[5]{6}$

(d) $\sqrt[5]{12}$

Given that, it can be simplified as

Therefore given expression is simplified and correct choice is.

#### Question 3:

The rationalisation factor of $\sqrt{3}$ is

(a) $-\sqrt{3}$

(b) $\frac{1}{\sqrt{3}}$

(c) $2\sqrt{3}$

(d) $-2\sqrt{3}$

We know that rationalization factor for is. Hence rationalization factor of is.Hence the correct option is.

#### Question 4:

The rationalisation factor of $2+\sqrt{3}$ is

(a) $2-\sqrt{3}$

(b) $2+\sqrt{3}$

(c) $\sqrt{2}-3$

(d) $\sqrt{3}-2$

We know that rationalization factor for is. Hence rationalization factor of is.Hence correct option is

#### Question 5:

If x = $\sqrt{5}+2$, then $x-\frac{1}{x}$ equals

(a) $2\sqrt{5}$

(b) 4

(c) 2

(d) $\sqrt{5}$

Given that.Hence is given as

.We need to find

We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Therefore,

Hence the correct option is.

#### Question 6:

If $\frac{\sqrt{3-1}}{\sqrt{3}+1}$ = $a-b\sqrt{3}$, then

(a) a = 2, b =1

(b) a = 2, b =−1

(c) a = −2, b = 1

(d) a = b = 1

Given that:

We are asked to find a and b

We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

On equating rational and irrational terms, we get

Comparing rational and irrational part we get

Hence, the correct choice is.

#### Question 7:

The simplest rationalising factor of $\sqrt[3]{500}$ is

(a) $\sqrt[3]{2}$

(b) $\sqrt[3]{5}$

(c) $\sqrt{3}$

(d) none of these

Given that:.To find simplest rationalizing factor of the given expression we will factorize it as

The rationalizing factor of is, since when we multiply given expression with this factor we get rid of irrational term.

Therefore, rationalizing factor of the given expression is

Hence correct option is.

#### Question 8:

The simplest rationalising factor of $\sqrt{3}+\sqrt{5}$ is

(a) $\sqrt{3}-5$

(b) $3-\sqrt{5}$

(c) $\sqrt{3}-\sqrt{5}$

(d) $\sqrt{3}+\sqrt{5}$

We know that rationalization factor for is. Hence rationalization factor of is.

#### Question 9:

The simplest rationalising factor of $2\sqrt{5}$$\sqrt{3}$ is

(a) $2\sqrt{5}+3$

(b) $2\sqrt{5}+\sqrt{3}$

(c) $\sqrt{5}+\sqrt{3}$

(d) $\sqrt{5}-\sqrt{3}$

We know that rationalization factor for is. Hence rationalization factor of is.

#### Question 10:

If x =$\frac{2}{3+\sqrt{7}}$, then (x−3)2 =

(a) 1
(b) 3
(c) 6
(d) 7

Given that:

We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Therefore,

On squaring both sides, we get

Hence the value of the given expression is.

#### Question 11:

If $x=7+4\sqrt{3}$ and xy =1, then $\frac{1}{{x}^{2}}+\frac{1}{{y}^{2}}=$

(a) 64

(b) 134

(c) 194

(d) 1/49

Given that,

Hence is given as

We need to find

We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Since so we have

Therefore,

Hence the value of the given expression is.

#### Question 12:

If then $x+\frac{1}{x}$=

(a) 2
(b) 4
(c) 8
(d) 1

Given that .It can be simplified as

We need to find

We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Therefore,

Hence the value of the given expression is 8.Hence correct option is .

#### Question 13:

If $x=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$ and $y=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$, then x + y +xy=

(a) 9
(b) 5
(c) 17
(d) 7

Given that and.

Now we will rationalize x. We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Similarly, we can rationalize y. We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Therefore,

Hence the value of the given expression is.

#### Question 14:

If x= and y = , then x2 + y +y2 =

(a) 101
(b) 99
(c) 98
(d) 102

Given that and.

We need to find

Now we will rationalize x. We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Similarly, we can rationalize y. We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Therefore,

Hence the value of the given expression is.

#### Question 15:

$\frac{1}{\sqrt{9}-\sqrt{8}}$ is equal to

(a) $3+2\sqrt{2}$

(b) $\frac{1}{3+2\sqrt{2}}$

(c) $3-2\sqrt{2}$

(d) $\frac{3}{2}-\sqrt{2}$

Given that

We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Hence the correct option is.

#### Question 16:

The value of $\frac{\sqrt{48}+\sqrt{32}}{\sqrt{27}+\sqrt{18}}$ is

(a) $\frac{4}{3}$

(b) 4

(c) 3

(d) $\frac{3}{4}$

Given that

We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

We can factor irrational terms as

Hence the value of given expression is.

#### Question 17:

If $\frac{5-\sqrt{3}}{2+\sqrt{3}}=x+y\sqrt{3}$, then

(a) x = 13, y = −7

(b) x = −13, y = 7

(c) x = −13, y = −7

(d) x = 13, y = 7

Given that: .We need to find x and y

We know that rationalization factor for is . We will multiply numerator and denominator of the given expression   by, to get

Since

On equating rational and irrational terms, we get

Hence, the correct choice is.

#### Question 18:

If x = $\sqrt[3]{2+\sqrt{3}}$, then ${x}^{3}+\frac{1}{{x}^{3}}=$

(a) 2

(b) 4

(c) 8

(d) 9

Given that .It can be simplified as

We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Therefore,

Hence the value of the given expression is.

#### Question 19:

The value of $\sqrt{3-2\sqrt{2}}$ is

(a) $\sqrt{2}-1$

(b) $\sqrt{2}+1$

(c) $\sqrt{3}-\sqrt{2}$

(d) $\sqrt{3}+\sqrt{2}$

Given that:.It can be written in the form as

Hence the value of the given expression is.

#### Question 20:

The value of is

(a) $\sqrt{3}-\sqrt{2}$

(b) $\sqrt{3}+\sqrt{2}$

(c) $\sqrt{5}+\sqrt{6}$

(d) none of these

Given that:.It can be written in the form as

Hence the value of the given expression is.

#### Question 21:

If $\sqrt{2}=1.4142$ then $\sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}}$ is equal to

(a) 0.1718

(b) 5.8282

(c) 0.4142

(d) 2.4142

Given that , we need to find the value of .

We can rationalize the denominator of the given expression. We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

$\sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}}=\frac{\sqrt{2}-1}{1}$

Putting the value of , we get

Hence the value of the given expression is 0.14142 and correct choice is.

#### Question 22:

If then the value of $\sqrt{6}-\sqrt{3}$ upto three places of decimal is

(a) 0.235

(b) 0.707

(c) 1.414

(d) 0.471

Given that.We need to find.

We can factor out from the given expression, to get

Putting the value of, we get

Hence the value of expression must closely resemble be

The correct option is.

#### Question 23:

The positive square root of $7+\sqrt{48}$ is

(a) $7+2\sqrt{3}$

(b) $7+\sqrt{3}$

(c) $2+\sqrt{3}$

(d) $3+\sqrt{2}$

Given that:.To find square root of the given expression we need to rewrite the expression in the form

Hence the square root of the given expression is

Hence the correct option is.

#### Question 24:

If $x=\sqrt{6}+\sqrt{5}$, then ${x}^{2}+\frac{1}{{x}^{2}}-2=$

(a) $2\sqrt{6}$

(b) $2\sqrt{5}$

(c) 24

(d) 20

Given that.Hence is given as

We need to find

We know that rationalization factor for   is. We will multiply numerator and denominator of the given expression   by, to get

We know that  therefore,

Hence the value of the given expression is 20 and correct option is (d).

#### Question 25:

If

(a) −5

(b) −6

(c) −4

(d) −2

Given that:

We need to find a

The given expression can be simplified by taking square on both sides

The irrational terms on right side can be factorized such that it of the same form as left side terms.

Hence,

On comparing rational and irrational terms, we get.Therefore, correct choice is .

#### Question 1:

The number obtained by rationalizing the denominator of $\frac{1}{\sqrt{7}+2}$ is __________.

Hence, the number obtained by rationalizing the denominator of $\frac{1}{\sqrt{7}+2}$ is .

#### Question 2:

If $\frac{1}{\sqrt{9}-\sqrt{8}}=A+B\sqrt{2}$, then A = ____________ and B = ____________.

Hence, if $\frac{1}{\sqrt{9}-\sqrt{8}}=A+B\sqrt{2}$, then A = 3 and B = 2.

#### Question 3:

After rationalizing the denominator of $\frac{7}{3\sqrt{3}-2\sqrt{2}}$, we get the denominator as __________.

Hence, after rationalizing the denominator of $\frac{7}{3\sqrt{3}-2\sqrt{2}}$, we get the denominator as $\overline{)\frac{21\sqrt{3}+14\sqrt{2}}{19}}.$

If

Hence, if

If

Hence, if

If

Hence, if

If

Hence, if

#### Question 8:

If , then x + y = __________.

Hence, x + y = $\overline{)-2\sqrt{3}}$.

If

Hence, if

#### Question 10:

$\sqrt{7+2\sqrt{6}}-\sqrt{7-2\sqrt{6}}=__________.$

Hence, $\sqrt{7+2\sqrt{6}}-\sqrt{7-2\sqrt{6}}=\overline{)2}.$

If

Hence, if

If

Hence, if

#### Question 1:

Write the value of

Given that

It can be simplified as

Hence the value of the given expression is.

#### Question 2:

Write the reciprocal of $5+\sqrt{2}$.

Given that, it’s reciprocal is given as

It can be simplified by rationalizing the denominator. The rationalizing factor of is, we will multiply numerator and denominator of the given expression by, to get

Hence reciprocal of the given expression is.

#### Question 3:

Write the rationalisation factor of $7-3\sqrt{5}$.

The rationalizing factor of is. Hence the rationalizing factor of is .

#### Question 4:

If find the values of x and y.

It is given that;

.we need to find x and y

We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

On equating rational and irrational terms, we get

Hence, we get.

#### Question 5:

If x=$\sqrt{2}-1$, then write the value of $\frac{1}{x}.\phantom{\rule{0ex}{0ex}}$

Given that.Hence is given as

We know that rationalization factor for is . We will multiply each side of the given expression by, to get

Hence the value of the given expression is.

#### Question 6:

If $a=\sqrt{2}+1$, then find the value of $a-\frac{1}{a}$.

Given that, hence is given as

We know that rationalization factor for is . We will multiply each side of the given expression by, to get

Therefore,

Hence value of the given expression is.

#### Question 7:

If $x=2+\sqrt{3}$,  find the value of $x+\frac{1}{x}$.

Given that, hence $\frac{1}{x}$is given as

We know that rationalization factor for is . We will multiply each side of the given expression by, to get

Therefore,

Hence value of the given expression is.

#### Question 8:

Write the rationalisation factor of $\sqrt{5}-2$.

Given that, we know that rationalization factor of is

So the rationalization factor of is.

#### Question 9:

Simplify $\sqrt{3+2\sqrt{2}}$.

We are asked to simplify. It can be written in the form as

Hence the value of given expression is.

#### Question 10:

Simplify $\sqrt{3-2\sqrt{2}}$.

We are asked to simplify. It can be written in the form as

Hence the value of the given expression is.

#### Question 11:

If , then find the value of $\sqrt{x}-\frac{1}{\sqrt{x}}$.

Given that:.It can be written in the form as

Therefore,

We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence,

Therefore, value of the given expression is.

#### Question 1:

Simplify each of the following:

(i) $\sqrt[3]{4}×\sqrt[3]{16}$

(ii) $\frac{\sqrt[4]{1250}}{\sqrt[4]{2}}$

(i) We know that. We will use this property to simplify the expression.

Hence the value of the given expression is .

(ii) We know that. We will use this property to simplify the expression.

Hence the value of the given expression is.

#### Question 2:

Simplify the following expressions:

(i)

(ii)

(iii)

(i) We can simplify the expression as

Hence the value of the expression is

(ii) We can simplify the expression as

Hence the value of the expression is

(iii) We can simplify the expression as

Hence the value of the expression is .

#### Question 3:

Simplify the following expressions:

(i)

(ii)

(iii)

(iv) $\left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right)$

(v)

(i) We know that. We will use this property to simplify the expression.

Hence the value of expression is 110.

(ii) We know that. We will use this property to simplify the expression.

Hence the value of expression is 18.

(iii) We know that. We will use this property to simplify the expression.

Hence the value of expression is 6

(iv) We know that. We will use this property to simplify the expression.

Hence the value of expression is 6.

(v) We know that. We will use this property to simplify the expression.

Hence the value of expression is 3.

#### Question 4:

Simplify the following expressions:

(i) ${\left(\sqrt{3}+\sqrt{7}\right)}^{2}$

(ii) ${\left(\sqrt{5}-\sqrt{3}\right)}^{2}$

(iii) ${\left(2\sqrt{5}+3\sqrt{2}\right)}^{2}$

(i) We know that. We will use this property to simplify the expression.

Hence the value of expression is

(ii) We know that. We will use this property to simplify the expression.

Hence the value of expression is

(iii) We know that. We will use this property to simplify the expression.

Hence the value of expression is.

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