Identities for Irrational Numbers

Q.

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

Q.

Simplify $\sqrt{72}+\sqrt{800}-\sqrt{18}$.

Q.

Simplify: $6\sqrt{36}+5\sqrt{12}$

Q.

Simplify $(2\sqrt{5}+3\sqrt{2}{)}^2$

Q.

$(\sqrt{a}-\sqrt{b})\times (\sqrt{a}+\sqrt{b})=$

1. $\sqrt{a}-b$

2. $a+\sqrt{b}$

3. $a-b$

4. $a+b$

Q.

$(-2-\sqrt{3})(-2+\sqrt{3})$ when simplified is

(a) positive and irrational (b) positive and rational (c) negative and irrational (d) negative and rational

Q.

Which of the following is the value of $(\sqrt{11}-\sqrt{7})(\sqrt{11}+\sqrt{7})$?

(a) -4

(b) 4

(c) $\sqrt{11}$

(d) $\sqrt{7}$

Q. The value of $(\sqrt{3}+\sqrt{2}{)}^2$ is equal to ____.

1. $5+2\sqrt{3}$
2. $5+2\sqrt{2}$
3. $5+2\sqrt{6}$
4. $3+2\sqrt{6}$

Q. Find the sum of the squares of the following: $\frac{\sqrt{3}}{\sqrt{2}+1},\frac{\sqrt{3}}{\sqrt{2}-1},\frac{\sqrt{2}}{\sqrt{3}}$

Q.

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

(a)$\sqrt{3}+\sqrt{2}$ (b)$\sqrt{3}-\sqrt{2}$ (c)$\sqrt{2}+1$ (d)$\sqrt{2}-1$

Q.

Solve $(3-\sqrt{11})(3+\sqrt{11})$

Q.

Question 2 (iv)
Simplify $(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})$ .

Q.

Find the values of $a$ and $b$

if $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=a+b\sqrt{3}$

1. $a=11,b=-6$

2. $a=9,b=-6$

3. $a=-11,b=6$

4. $a=-9,b=6$

Q.

Show all work to multiply quantity $6$ plus the square root of negative $64$ end quantity times quantity $3$ minus the square root of negative $16$ end quantity.

Q.

On her birthday Reema distributed chocolates in an orphanage. The total number of chocolates she distributed is given by $(5+\sqrt{11})(5-\sqrt{11})$

Q. The value of $(\sqrt{7}+\sqrt{6})(\sqrt{7}-\sqrt{6})$ is equal to

1. 3
2. 1
3. 8
4. 13

Q.

Question 2 (iv)
Simplify $(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})$ .

Q.

Simplify

(i) $(3-\sqrt{11})(3+\sqrt{11})$

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

(iii) $(3-\sqrt{3}{)}^2$

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

(v) $(5+\sqrt{7})(2+\sqrt{5})$

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

Q. $\sqrt{3+2\sqrt{2}}-\sqrt{3-2\sqrt{2}}$ is equal to

1. $2$
2. $1$
3. $2\sqrt{2}$
4. $\sqrt{6}$

Q. The positive square root of $(\sqrt{48}-\sqrt{45})$ is _________.

1. $\frac{\sqrt[4]{43}}{\sqrt{2}}(\sqrt{5}-\sqrt{3})$
2. $\frac{\sqrt[4]{43}}{2}(\sqrt{5}-\sqrt{3})$
3. $\frac{\sqrt{2}}{\sqrt[4]{43}}(\sqrt{5}-\sqrt{3})$
4. $\frac{\sqrt[4]{43}}{\sqrt{2}}(\sqrt{5}+\sqrt{3})$

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

Q.

$\text{If}{x}^2=7-\sqrt{48}$, then $x$ = _______________

1. $2+\sqrt{3}$

2. $2-\sqrt{3}$

3. $2\sqrt{3}$

4. 4

Q.

$\sqrt[5]{56}\times \sqrt[5]{56}$ is equal to

1. $\sqrt[5]{56\times 0}$

2. $\sqrt[5]{56}$

3. $\sqrt[5]{512}$

4. $\sqrt[5]{536}$

Q. If $y=\sqrt{\sqrt{7}+7+\sqrt{8+2\sqrt{7}}}-\sqrt{7}$, then find the value of $y$.

1. 1
2. $\sqrt{7}$
3. $8-2\sqrt{7}$
4. $\sqrt{7}-1$

Q. Solve:
$\frac{\sqrt{24}}{8}+\frac{\sqrt{54}}{9}$.

Q. Solve for $x$ : $\sqrt{7x^2}-6x-13\sqrt{7}=0$

Q. ${64}^{-\frac{1}{3}}[{64}^{\frac{1}{3}}-{64}^{\frac{2}{3}}]=$ .

1. $\frac{1}{3}$
2. $-3$
3. $\frac{2}{3}$

Q. Solve: $5\sqrt{5}{x}^2+20x+3\sqrt{5}=0$

Q. If the square root of $\sqrt{35-12\sqrt{6}-4\sqrt{15}+6\sqrt{10}}$ = $m\sqrt{2}-k\sqrt{3}+\sqrt{5}$ . Find k+m

Q. The value of $(\sqrt{3}+\sqrt{2}{)}^2$ is equal to ____.

1. $5+2\sqrt{3}$
2. $5+2\sqrt{2}$
3. $5+2\sqrt{6}$
4. $3+2\sqrt{6}$