Tag: numbers and place value

Questions Related to numbers and place value

Which of the following expressions is true?

maths numbers and place value many forms of ten thousand standard form of numbers number names, numerals and place values

  1. $2940000=$$\displaystyle 2\cdot 94\times 10^{5}$

  2. $502000=$$\displaystyle 5\cdot 02\times 10^{5}$

  3. $3683000=$$\displaystyle 3\cdot 683\times 10^{5}$

  4. $40404000=$$\displaystyle 4\cdot 0404\times 10^{5}$

Show answer
Correct Option: B
Explanation:

$\displaystyle 502000= 5\cdot 02\times 10^{5}$

Therefore, option B is correct.

When $70, 000$ is written as $7.0\times10^n$, what is the value of $n$?

maths numbers and place value many forms of ten thousand standard form of numbers number names, numerals and place values

  1. $1$

  2. $2$

  3. $3$

  4. $4$

  5. $5$

Show answer
Correct Option: D
Explanation:

Given that $70,000$ is written as $7.0$ $\times $ ${10}^{n}$

From this, we can write
$70,000$ $=$ $7.0$ $\times$ ${10}^{n}$
$\Rightarrow {10}^{n}$ $=$ $\dfrac {70,000}{7}$
$\Rightarrow {10}^{n}$ $=$ $10,000$
$\Rightarrow {10}^{n}$ $=$ ${10}^{4}$
$\Rightarrow n$ $=$ $\log _{10}$ ${10}^{4}$
$\Rightarrow n$ $=$ $4$ $\log _{10}$ $10$
$\Rightarrow $ $=$ $4$
Therefore, the value of $n$ is $'4'$.

The standard form of $15240000$ is __________.

maths numbers and place value many forms of ten thousand standard form of numbers number names, numerals and place values

  1. $1.524\times 10^{7}$

  2. $1.524\times 10^{6}$

  3. $15.24\times 10^{7}$

  4. $1.524\times 10^{8}$

Show answer
Correct Option: A
Explanation:

$15240000 = 1524\times 10000$

                   $= 1.524 \times 1000 \times 10000$
                   $= 1.524 \times 10^{7}$

Which of the following is equivalent to $ 7.7 \times 10^{-6}$?

maths numbers and place value many forms of ten thousand standard form of numbers number names, numerals and place values

  1. $0.00000077$

  2. $0.0000077$

  3. $0.000077$

  4. $0.00077$

Show answer
Correct Option: B
Explanation:

$7.7 \times 10^{-6}=\dfrac{7.7}{1000000}=0.0000077$

The number $3.02 \times10^{-6}$ can be expressed in decimal form as:

maths numbers and place value many forms of ten thousand standard form of numbers number names, numerals and place values

  1. $0.0000302$

  2. $0.00000302$

  3. $0.000302$

  4. $0.00302$

Show answer
Correct Option: B
Explanation:

$3.02 \times10^{-6}$ = $\dfrac{3.02}{1000000}$ $= 0.00000302$

The number $3\times10^{-8}$ can also be expressed as:

maths numbers and place value many forms of ten thousand standard form of numbers number names, numerals and place values

  1. $0.000003$

  2. $0.00003$

  3. $0.00000003$

  4. $0.0003$

Show answer
Correct Option: C
Explanation:

$3\times10^{-8}$ $=$ $\dfrac{3}{100000000}$ $=$ ${0.00000003}$

The value of $\dfrac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}$ is?

maths numbers and place value many forms of ten thousand standard form of numbers number names, numerals and place values

  1. $324$

  2. $400$

  3. $373$

  4. $1024$

Show answer
Correct Option: A

Find the unit digit of ${3^{46}} + 125 \times 436 + 256 \times {7^{345}}$

maths numbers and place value many forms of ten thousand standard form of numbers number names, numerals and place values

  1. $1$

  2. $3$

  3. $7$

  4. $9$

Show answer
Correct Option: B

Find the last two digits of $3^{1997}$.

maths numbers and place value many forms of ten thousand standard form of numbers number names, numerals and place values

  1. $67$

  2. $63$

  3. $80$

  4. $56$

Show answer
Correct Option: B
Explanation:

This is same as asking what is remainder when $3^{1997}\div 100$
$3^{4}\equiv 81  mod  100$
$3^{8}\equiv 61  mod  100$
$3^{12}\equiv 41  mod  100$
$3^{16}\equiv 21  mod  100$
$3^{20}\equiv 1  mod  100$


Now, $3^{40}, 3^{60}, 3^{80}, 3^{100}, ...., 3^{1980}$ all are $\equiv 1  mod  100$

We know $3^{16}\equiv 21  mod  100$

$3^{17}\equiv 21\times 3  mod  100$

$3^{17}\equiv 63  mod  100$

$\therefore 3^{1997}\equiv 3^{1980}\times 3^{17}$

since, $3^{1980}\equiv 1  mod  100$

and $3^{17}\equiv 63  mod  100$

$\therefore 3^{1997}\equiv 63  mod  100$

$\therefore $ Last two digit is 63

The value of ${\left( {{{27}^{\tfrac{{ - 2}}{3}}}} \right)^{\tfrac{1}{2}}} \times {\left( {{{64}^{\tfrac{1}{3}}}} \right)^2} \times {\left( {{{81}^{\tfrac{{ - 3}}{2}}}} \right)^{\tfrac{1}{6}}}$

maths numbers and place value many forms of ten thousand standard form of numbers number names, numerals and place values

  1. $\dfrac{1}{9}$

  2. $\dfrac{16}{9}$

  3. $\dfrac{2}{9}$

  4. $-\dfrac{16}{9}$

Show answer
Correct Option: B
Explanation:
${\left( {{{27}^{\tfrac{{ - 2}}{3}}}} \right)^{\tfrac{1}{2}}} \times {\left( {{{64}^{\tfrac{1}{3}}}} \right)^2} \times {\left( {{{81}^{\tfrac{{ - 3}}{2}}}} \right)^{\tfrac{1}{6}}}$

$\displaystyle =\left(\dfrac{1}{(27)^{\tfrac{2}{3}}}\right)^{\tfrac{1}{2}}\times (4^{3\times \tfrac{1}{3}})^{2}\times \left(\dfrac{1}{(81)^{\frac{3}{2}}}\right)^{\tfrac{1}{6}}$

$\displaystyle =\left(\dfrac{1}{27}\right)^{\tfrac{2}{3}\times \tfrac{1}{2}}\times (4)^{2}\times \left(\dfrac{1}{81}\right)^{\tfrac{3}{2}\times \tfrac{1}{6}}$

$\displaystyle =\left(\dfrac{1}{3^{3}}\right)^{\tfrac{1}{3}}\times (4)^{2}\times \left(\dfrac{1}{3^{4}}\right)^{\tfrac{1}{4}}$

$\displaystyle =\frac{1}{3}\times 16\times \frac{1}{3}$

$=\dfrac{16}{9}$