Introduction to Surds

Q.

Sum of infinity of the series $1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+\cdots$ is

1. $\frac{7}{16}$

2. $\frac{5}{16}$

3. $\frac{104}{64}$

4. $\frac{35}{16}$

Q.

The sum of infinity of the series $1+\frac{2}{3}+\frac{6}{3^2}+\frac{10}{3^3}+\frac{14}{3^4}+...$ is

1. $3$

2. $4$

3. $6$

4. $2$

Q.

If $1/1^4+1/2^4+1/3^4+1/4^4+...\infty ={\pi }^4/90,$ then the value of $\mathbf{1}\mathbf{/}{\mathbf{1}}^{\mathbf{4}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{1}\mathbf{/}{\mathbf{3}}^{\mathbf{4}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{1}\mathbf{/}{\mathbf{5}}^{\mathbf{4}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{\infty }$ is

1. ${\pi }^4/96$

2. ${\pi }^4/45$

3. $89{\pi }^4/90$

4. None of these

Q. $\sqrt{36}$ is a surd.

1. False
2. True

Q.

The sum of the series $\frac{3}{4}+\frac{5}{36}+\frac{7}{144}+...............$ upto $11$ terms is

1. $\frac{120}{121}$

2. $\frac{143}{144}$

3. $1$

4. $\frac{144}{143}$

Q.

Which of the following is not equal to $1$?

1. $\frac{\left(2^3\times 3^3\right)}{\left(4\times 18\right)}$

2. $\frac{\left[{\left(-2\right)}^3\times {\left(-2\right)}^4\right]}{{\left(-2\right)}^7}$

3. $\frac{3^0\times 5^3}{5\times 25}$

4. $\frac{2^4}{{\left(7^0+3^0\right)}^3}$

Q.

Which of the following is equal to ${(-\frac{3}{4})}^{-3}$?

1. ${\left(\frac{3}{4}\right)}^{-3}$

2. $-{\left(\frac{3}{4}\right)}^{-3}$

3. ${\left(\frac{4}{3}\right)}^3$

4. ${(-\frac{4}{3})}^3$

Q. Comment on the statement:
All surds are irrational numbers, but all irrational numbers are not surds.

1. The statement is correct.
2. The statement is wrong.
3. Cannot be determined.

Q. The $n^{th}$ root of any positive number $a$ can be written as

1. $a^n$
2. $a^{1/n}$
3. $\sqrt[n]{n{a}^n}$
4. $n\times a$

Q.

Evaluate the expression $\frac{1}{4}-2\sqrt[3]{-\frac{1}{216}}$.

Q.

If $a$ be any real number, then $\frac{a^2}{1+a^4}$

1. $\in (0,1)$

2. $\in (0,\frac{1}{2}]$

3. $\in (0,1]$

4. $[0,1]$

Q. What is the side length of a cube with a volume of $128m^3.$ Is the side of the cube a surd?

1. ${128}^{\frac{1}{3}}m$, yes
2. $\frac{128}{3}m$, no
3. ${128}^{3}m$, yes
4. ${128}^{\frac{1}{2}}m$, no

Q.

Simplify the following

$(221p^2{q}^2+13p{q}^2+17p^{2}q)÷\left(-663pq\right)$

Q. Select all the numbers that are surds:

1. $\sqrt[4]{481}$
2. $\sqrt[5]{581}$
3. $\sqrt{81}$
4. $\sqrt[7]{749}$
5. $\sqrt[5]{525}$
6. $\sqrt{25}$
7. $\sqrt[6]{626}$
8. $\sqrt[9]{99}$

Q.

Evaluate the expression ${\left(\sqrt[3]{-\frac{1}{8}}\right)}^3+\frac{3}{4}$ .

Q. $8000\sqrt[5]{58}{\text{cm}}^3$ of milk is to be fully filled in a container having equal dimensions. Calculate the height of the container.

1. $20\sqrt[5]{52}{\text{cm}}^3$
2. $20\sqrt[5]{52}\text{cm}$
3. $2\sqrt[5]{520}\text{cm}$

Q.

The simplest form of $\sqrt[5]{32}$ is .

Q. Choose the correct statement(s).

1. Any number that can be expressed as $\sqrt[n]{na}$ form, is a surd.
2. Any number that can be expressed as $\frac{p}{q}$ form, where $p$ and $q$ are integers and $q\ne 0,$ is irrational.
3. Euler's number ${}^{\prime }{e}^{\prime }$ is an irrational number.

Q. Choose the most suitable option for Surd.

1. Irrational number that cannot be written as $\frac{p}{q}$, where $p,q$ are integers and $q$ cannot be $0$
2. Irrational number that can be written as $\frac{p}{q}$, where $p,q$ are integers and $q$ cannot be $0$
3. Rational number that cannot be written as $\frac{p}{q}$, where $p,q$ are integers and $q$ cannot be $0$
4. Rational number that can be written as $\frac{p}{q}$, where $p,q$ are integers and $q$ cannot be $0$

Q. Determine the side of a square whose area is $4\sqrt{125}{\text{unit}}^2.$

1. $2\left(5\sqrt{5}\right)\text{unit}$
2. $2(5\sqrt{5}{)}^{\frac{1}{2}}\text{unit}$
3. $5(5\sqrt{5}{)}^{\frac{1}{2}}\text{unit}$
4. $2(5\sqrt{5}{)}^{\frac{1}{2}}{\text{unit}}^2$

Q. Simplify the expression:
$\frac{\sqrt{20}+\frac{\sqrt{10}}{\sqrt{18}}}{5^{\frac{1}{2}}}$

Q.

If $\frac{\mathrm{p}}{\mathrm{q}}={\left(\frac{3}{4}\right)}^{18}÷{\left(\frac{3}{4}\right)}^{16}$, find the value of ${\left(\frac{\mathrm{p}}{\mathrm{q}}\right)}^2$.

Q.

Simplify:
$\left(39÷3\right)+7$

Q.

Simplify:

$\frac{4}{5}\mathrm{of}\frac{205}{64}\times 7\left[\sqrt{256}÷32\left\{77+\overline{12-3}\left(\sqrt{49}+5\right)\right\}\right]$