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Standard form of numbers - class-VIII

Description: standard form of numbers
Number of Questions: 36
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Tags: number world exponents numbers and place value maths place value, ordering and rounding powers and exponents
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In standard from, the number 829030000 is written as $K\times { 10 }^{ 8 }$ where K is equal to 

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

  1. 82903

  2. 829.03

  3. 82.903

  4. 8.2903

Show answer
Correct Option: A
Explanation:

$829030000=K \times 10^8$

Thus, $K=8.2903$
Hence, option D is correct.

The sum of the powers of the prime factors in $108 \times 192$  is

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

  1. $5$

  2. $7$

  3. $8$

  4. $12$

Show answer
Correct Option: D
Explanation:

$\displaystyle 108=2\times 2\times  3\times 3\times 3=2^{2}\times 3^{3}$
$\displaystyle 192=2\times2\times2\times2\times2\times2\times3$
$\displaystyle =2^{6}\times 3^{1}$
$\displaystyle 108 \times 192=2^{2}\times 3^{3}\times 2^{6}\times 3^{1}$
$\displaystyle =2^{8}\times 3^{4}$
Sum of the powers = 8 + 4 = 12

The value of $\displaystyle (243)^{\frac{-2}{5}}$ is---

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

  1. $\displaystyle \frac{1}{9}$

  2. $\displaystyle \frac{2}{9}$

  3. 9

  4. None of these

Show answer
Correct Option: A
Explanation:

$\displaystyle (243)^{-2/5} =\displaystyle(3^5)^{-2/5}$
                $\displaystyle = 3^{-2} = \frac{1}{9}$

Find the value of $\displaystyle (64)^{-2/3}$---

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

  1. 16

  2. $\displaystyle \frac{1}{16}$

  3. $\displaystyle -\frac{1}{16}$

  4. None of these

Show answer
Correct Option: B
Explanation:

$\displaystyle (64)^{-2/3} = 4^3 \times 1^{-2/3} = 4^{-2}$
                 $\displaystyle = \frac{1}{4^2} = \frac{1}{16}$   

The value of $[(-3)^{(-2)}]^{(-3)}$ is---

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

  1. 243

  2. 27

  3. 729

  4. None of these

Show answer
Correct Option: C
Explanation:

$[(-3)^{(-2)}]^{(-3)} = (-3)^6$
                           = 729

Charge of an electron is $0.00000000000000000016$ coulomb. This number can also be written in standard form as:

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

  1. $\displaystyle 1\cdot 6\times 10^{19}$

  2. $\displaystyle 1\cdot 6\times 10^{-20}$

  3. $\displaystyle 1\cdot 6\times 10^{-19}$

  4. $\displaystyle 1\cdot 6\times 10^{18}$

Show answer
Correct Option: C
Explanation:

$0.0000000000000000000016=\displaystyle \frac{16}{100000000000000000000}$


=$\displaystyle \frac{1\cdot 6\times 10^{1}}{10^{20}}$

=$\displaystyle 1\cdot 6\times 10^{-19}$

The value of $(3^0 - 2^1) \times 4^2$ is---

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

  1. -16

  2. 32

  3. 64

  4. 0

Show answer
Correct Option: A
Explanation:

$(3^0 - 2^1) \times 4^2 = (1-2) \times 16$
                                         $= -1 \times 16 = -16$

Simplify $\displaystyle (27)^{\frac{-2}{3}} \div \displaystyle (64)^{\frac{-2}{3}}$ is---

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

  1. $\displaystyle \frac{9}{16}$

  2. 16

  3. $\displaystyle \frac{16}{9}$

  4. 9

Show answer
Correct Option: C
Explanation:

$\displaystyle \frac{(27)^{-2/3}}{ ( 64)^{-2/3}} = \frac{\displaystyle \frac{1}{9}}{ \displaystyle \frac{14}{16}} = \frac{16}{9}$

Size of a bacteria is $\displaystyle 1.5\times 10^{-7}m$. This number can also be written as:

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

  1. $0.00000015$

  2. $0.0000015$

  3. $0.000000015$

  4. $15000000$

Show answer
Correct Option: A
Explanation:
$a\times 10^{-k} = \dfrac{a}{10^k}$

$\displaystyle 1.5 \times 10^{-7}= \dfrac{1.5}{10^7} = 0.00000015$

The usual form of $\displaystyle 6\cdot 8793\times 10^{4}$ is:

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

  1. $687930$

  2. $68793$

  3. $6879.3$

  4. $6879300$

Show answer
Correct Option: B
Explanation:

$\displaystyle 6\cdot 8793\times 10^{4}=68,793$

Which of the following statement is false?

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

  1. $\displaystyle 4\cdot 59\times 10^{-3}=0.00459$

  2. $\displaystyle 7\times 10^{-5}=0.00007$

  3. $\displaystyle 1\cdot 03\times 10^{-3}=1030$

  4. $\displaystyle 8\cdot 8\times 10^{-4}=0.00088$

Show answer
Correct Option: C
Explanation:

$\displaystyle 1\cdot 03\times 10^{-3}= 0\cdot 00103$
$\displaystyle \therefore $ The given statement is false

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

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

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

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

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

$=\dfrac{16}{9}$

The distance of the earth from the sun is 149,000,000 km. In scientific notation the distance is:

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

  1. $\displaystyle 149\times 10^{6}$ km

  2. $\displaystyle 14.9\times 10^{7}$ km

  3. $\displaystyle 1.49\times 10^{8}$ km

  4. $\displaystyle 0.149\times 10^{9}$ km

Show answer
Correct Option: C
Explanation:

$\displaystyle 149,000,000 km=149\times1000000=1.49\times100000000=1.49\times 10^{8}km$

Solve:

$ \displaystyle 9^{\dfrac{3}{2}\div (243)^{-\dfrac{2}{3}}} $  simplifies to

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

  1. $ \displaystyle 3^{\dfrac{10}{3}} $

  2. ${{3}^{{{3}^{\dfrac{13}{3}}}}} $

  3. $ \displaystyle 3^{\dfrac{1}{3}} $

  4. $ \displaystyle 3^{19} $

Show answer
Correct Option: B
Explanation:

Consider the given expression,


  $ \Rightarrow {{9}^{\dfrac{3}{2}\div {{\left( 243 \right)}^{-\,\dfrac{2}{3}}}}}={{9}^{\dfrac{3}{2}\times {{\left( 243 \right)}^{\dfrac{2}{3}}}}} $

 $ ={{9}^{\dfrac{3}{2}\times {{\left( {{3}^{5}} \right)}^{\dfrac{2}{3}}}}}={{9}^{\dfrac{3}{2}\times {{3}^{\dfrac{10}{3}}}}} $

 $ ={{\left( {{3}^{2}} \right)}^{\dfrac{3}{2}\times {{3}^{\dfrac{10}{3}}}}}={{\left( 3 \right)}^{3\times {{3}^{\dfrac{10}{3}}}}} $

 $ ={{3}^{{{3}^{1+\dfrac{10}{3}}}}}={{3}^{{{3}^{\dfrac{13}{3}}}}} $


Hence, this is the answer. 

The standard form of $0.000000000000487$ is ______

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

  1. $\displaystyle 4\cdot 87\times 10^{-13}$

  2. $\displaystyle 4\cdot 87\times 10^{-14}$

  3. $\displaystyle 4\cdot 87\times 10^{-15}$

  4. $\displaystyle 4\cdot 87\times 10^{-12}$

Show answer
Correct Option: A
Explanation:

$0.000000000000487=\displaystyle \frac{487}{1000000000000000}$

=$ \dfrac{487}{10^{15}}$ =$\displaystyle \frac{4\cdot 87\times 10^{2}}{10^{15}}$
=$\displaystyle 4\cdot 87\times 10^{-13}$

The standard form of $8,60,00,00,00,00,000$ is

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

  1. $\displaystyle 8\cdot 6\times 10^{13}$

  2. $\displaystyle 8\cdot 6\times 10^{-13}$

  3. $\displaystyle 8\cdot 6\times 10^{12}$

  4. $\displaystyle 8\cdot 6\times 10^{14}$

Show answer
Correct Option: A
Explanation:

$8,60,00,00,00,00,000=$$\displaystyle 8\cdot 6\times 10^{13}$

A number is said to be in the standard form when it is written as $\displaystyle k\times 10^{n}$ where $n$ is an integer and:

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

  1. $\displaystyle 1< k< 10$

  2. $\displaystyle 1< k\leqslant 10$

  3. $\displaystyle 1\leqslant k< 10$

  4. $\displaystyle 1\leqslant k\leqslant 10$

Show answer
Correct Option: C
Explanation:

$k\times10^n$


If $n$ is an integer, then $k$ should lie between $1$ and $10$.

Lets consider the equalties, $k$ cannot be equal to $10$, if it is then we can update $k = 1$ and increment $n$ by $1$.

So, $1\leq k <10$

The usual form of $\displaystyle 5\times 10^{-8}$ is

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

  1. $0.000005$

  2. $0.00000005$

  3. $0.0000005$

  4. $0.000000005$

Show answer
Correct Option: B
Explanation:

$\displaystyle 5\times 10^{-8}=0\cdot 00000005$

The usual form of $\displaystyle 4\cdot 56\times 10^{-5}$ is:

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

  1. $0.0000456$

  2. $0.00000456$

  3. $0.000456$

  4. $456000$

Show answer
Correct Option: A
Explanation:

$\displaystyle 4\cdot 56\times 10^{-5}=0\cdot 00000456$

If $0.00044=$$\displaystyle 4\cdot 4\times 10^{n}$ then, find 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. $4$

  2. $5$

  3. $-4$

  4. $-5$

Show answer
Correct Option: C
Explanation:

$\displaystyle 0\cdot 00044= \frac{44}{100000}= \frac{4\cdot 4\times10^{1}}{10^{5}}$
$=\displaystyle 4\cdot 4\times 10^{-4}$
Now
$\displaystyle 4\cdot 4\times 10^{-4}=4\cdot 4\times 10^{n}$
$\displaystyle \therefore n=-4$

Find the value of $n$ such that $502000000=$$\displaystyle 5\cdot 02\times 10^{n}$.

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

  1. $5$

  2. $7$

  3. $-8$

  4. $8$

Show answer
Correct Option: D
Explanation:

$\displaystyle 50,20,00,000=5\cdot 02\times 10^{8}$
Now $\displaystyle 5\cdot 02\times 10^{8}= 5\cdot 02\times 10^{n}$
$\displaystyle \therefore n= 8$

The standard form of $\displaystyle \frac{1}{10000000}$ is:

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

  1. $\displaystyle 1\times 10^{-7}$

  2. $\displaystyle 0\cdot 1\times 10^{-7}$

  3. $\displaystyle 1\times 10^{7}$

  4. $\displaystyle 1\times 10^{-6}$

Show answer
Correct Option: A
Explanation:

$\displaystyle \frac{1}{10000000}= \frac{1}{10^{7}}= 1\times 10^{-7}$

The distance of the sun from the earth is $1,49,60,00,00,000$ m. Express it in standard form.

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

  1. $\displaystyle 1\cdot 496\times 10^{-10}$

  2. $\displaystyle 1\cdot 496\times 10^{-11}$

  3. $\displaystyle 1\cdot 496\times 10^{10}$

  4. $\displaystyle 1\cdot 496\times 10^{11}$

Show answer
Correct Option: D
Explanation:

$1,49,60,00,00,000=$$\displaystyle 1\cdot 496\times 10^{11}$ m

The size of a plant cell is $0.00005473$ m. This number can also be written as

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

  1. $\displaystyle 5\cdot 473\times 10^{-3}$

  2. $\displaystyle 5\cdot 473\times 10^{-5}$

  3. $\displaystyle 5\cdot 473\times 10^{11}$

  4. $\displaystyle 5\cdot 473\times 10^{5}$

Show answer
Correct Option: B
Explanation:

$\displaystyle 0\cdot 00005473=\frac{5473}{100000000}=\frac{5\cdot 473\times 10^{3}}{10^{8}}$
=$\displaystyle 5\cdot 473\times 10^{-5}$

The usual form of $\displaystyle 2\cdot 73\times 10^{12}$ is:

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

  1. $273000000000$

  2. $2.730000000000$

  3. $2730000000000$

  4. $27300000000000$

Show answer
Correct Option: C
Explanation:

$2.73 \times 10^{12} $$=2730000000000$

The value of ${ \left( 256 \right)  }^{ 0.16 }.{ \left( 256 \right)  }^{ 0.09 }$ is:

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

  1. $4$

  2. $16$

  3. $64$

  4. $256.25$

  5. $-16$

Show answer
Correct Option: A
Explanation:

we know,


$a^{m} \times a^{n}=a^{m+n}$

and $256=4^{4}$

$\Rightarrow (256)^{0.16} \times (256)^{0.09}$ $=(256)^{0.25}$

we know,
$(a^{m})^{n}=a^{mn}$

$\Rightarrow (4^{4})^{0.25}$$=4^{4 \times 0.25}$$=4^1=4$

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