Scientific Notation Questions with Solutions

Questions on scientific notation are presented along with answers and detailed solutions .
Scientific notation of a number is of form: \( a \times 10^n \) where \( 1 \le a \lt 10 \), \( n \) is an integer and \( a \) is called the mantissa.


  1.   Which of the following is not in scientific notation?
    \( a: \; \; 23.7 \times 10^6 \) ,   \( b: \; \; - 2.31 \times 10^6 \) ,   \( c: \; \; - 2.3 \times 5^6 \) ,   \( d: \; \; 0.3 \times 10^{-4} \) ,   \( e: \; \; 10.0 \times 10^{2} \)
    A) \( \quad a , c \) and \( d\) only
    B) \(\quad c \) only
    C) \( \quad a \) and \( e \) only
    D) \( \quad c \) and \( e \) only
    E) \( \quad a , c , d \) and \( e \)

  2.   Which of the following scientific notations is equivalent to \( 510000 \)?
    A) \( \quad 5.1 \times 10^4 \)
    B) \( \quad 5.1 \times 10^{-5} \)
    C) \( \quad5.1 \times 10^6 \)
    D) \( \quad 5.1 \times 10^{-6} \)
    E) \( \quad 5.1 \times 10^5 \)

  3.   In scientific notation \( 100000 + 3000000 = \)
    A) \( \quad 3.1 \times 10^6 \)
    B) \( \quad 3.1 \times 10^7 \)
    C) \( \quad 3.1 \times 10^8 \)
    D) \( \quad 3.0 \times 10^6 \)
    E) \( \quad 3.1 \times 10^7 \)

  4.   Which of the following scientific notation is equivalent to \( 0.0000028\)?
    A) \( \quad 2.8 \times 10^6 \)
    B) \( \quad 2.8 \times 10^{-5} \)
    C) \( \quad 2.8 \times 10^{-6} \)
    D) \( \quad 2.8 \times 10^{-4} \)
    E) \( \quad 2.8 \times 10^5 \)

  5.   \( 1.2 \times 10^6 = \)
    A) \( \quad 120000 \)
    B) \( \quad 12000000 \)
    C) \( \quad 12000 \)
    D) \( \quad 1200000 \)
    E) \( \quad 0.0000012 \)

  6.   \( 2.3 \times 10^{-5} = \)
    A) \( \quad 230000 \)
    B) \( \quad 0.000023 \)
    C) \( \quad 0.00023 \)
    D) \( \quad 0.0023 \)
    E) \( \quad 2300000 \)

  7.   In scientific notation \( \dfrac{1}{10000}+\dfrac{2}{1000000} = \)
    A) \( \quad 1.02 \times 10^{-5} \)
    B) \( \quad 1.0 \times 10^{-4} \)
    C) \( \quad 1.02 \times 10^{-6} \)
    D) \( \quad 1.0 \times 10^{-5} \)
    E) \( \quad 1.02 \times 10^{-4} \)

  8.  In scientific notation, \( 2.1 \times 10^{-5} + 3.0 \times 10^{-4} = \)
    A) \( \quad 3.21 \times 10^{-4} \)
    B) \( \quad 3.21 \times 10^{-3} \)
    C) \( \quad 3.21 \times 10^{-5} \)
    D) \( \quad 3.21 \times 10^{-6} \)
    E) \( \quad 3.21 \times 10^{-2} \)

  9.   In scientific notation \( 120.054 \times 10^{-6} = \)
    A) \( \quad 1.20054 \times 10^{-6} \)
    B) \( \quad 1.20054 \times 10^{-8} \)
    C) \( \quad 1.20054 \times 10^{-4} \)
    D) \( \quad 1.20054 \times 10^{-6} \)
    E) \( \quad 120.054 \times 10^{-4} \)

  10.   \( 5.9 \times 10^{-6} - 2.0 \times 10^{-7} = \)
    A) \( \quad 5.7 \times 10^{-7} \)
    B) \( \quad 5.7 \times 10^{-6} \)
    C) \( \quad 3.9 \times 10^{-6} \)
    D) \( \quad 7.9 \times 10^{-6} \)
    E) \( \quad 5.7 \times 10^{-5} \)

Solutions to the Above Questions

  1.  Answer E
    Solution
    In \( a \), the mantissa \( 23.7 \) is greater than \(10 \), therefore it is not in scientific notation.
    In \( c \), the base \( 5 \) must be base \( 10 \).
    In \( d \), the mantissa \( 0.3 \) is less than \(1 \), therefore it is not in scientific notation.
    In \( e \), the mantissa \( 10.0 \) is equal to \(10 \), therefore it is not in scientific notation.

  2.  Answer E
    Solution
    Scientific notation of a number is of form: \( a \times 10^n \) where \( 1 \le a \lt 10 \) and \( n \) is an integer.
    \( 510000 \) is bigger than \( 10 \), therefore start with a decimal point from the right \( 510000. \) and move it till the number is between \( 1 \) and \( 10 \) not included.
    Hence for 510000, if we start with a decimal point from the right, we need to move the decimal point n = 5 times in order to obtain \( 5.10000 \) which is a number between \( 1 \) and \( 10 \) not included.
    \( 510,000 \) is written in scientific notation as: \( 5.1 \times 10^{5} \)

  3.  Answer A
    Solution
    We first add the numbers: \( 100000 + 3000000 = 3100000 \)
    \( 3100000 \) is bigger than 10, we therefore start with a decimal point on the right and move it \( n = 6 \) times to obtain \( 3.1 \) which is a number between \( 1 \) and \( 10 \) not included.
    Hence \( 100000 + 3,000000 = 3.1 \times 10^{6} \)

  4.  Answer C
    Solution
    The given number \( 0.0000028 \) is smaller than one and in order to obtain a number between \( 1 \) and \( 10 \) not included, we need to move the decimal point to the RIGHT.
    We need to move the decimal point \( n = 6 \) times in order to write the given number as \( 2.8 \), between \( 1 \) and \( 10 \) not included
    Hence \( 0.0000028 \) is written in scientific notation as: \( 2.8 \times 10^{-6} \)

  5.  Answer D
    Solution
    In this question, we are given the scientific notation of a number and asked to write it in standard form. Hence
    \( 1.2 \times 10^6 = 1.2 \times 1000000 = 1200000 \)

  6.  Answer B
    Solution
    We are given the scientific notation: \( 2.3 \times 10^{-5} = \dfrac{2.3}{100000} = 0.000023 \)

  7.  Answer E
    Solution
    Write in decimal form: \( \quad \dfrac{1}{10000}+\dfrac{2}{1000000} = 0.0001 + 0.000002 = 0.000102 \)

    In scientific notation: \( \quad \dfrac{1}{10000}+\dfrac{2}{1000000} = 0.000102 = 1.02 \times 10^{-4} \)

  8.  Answer A
    Solution
    Write in decimal form: \( \quad 2.1 \times 10^{-5} + 3.0 \times 10^{-4} = 0.000021 + 0.0003 = 0.000321 \)
    In scientific notation: \( \quad 2.1 \times 10^{-5} + 3.0 \times 10^{-4} = 0.000021 + 0.0003 = 0.000321 = 3.21 \times 10^{-4} \)

  9.  Answer C
    Solution
    The mantissan \( 120.054 \) is larger than \( 1 \), hence we need to move the decimal point \( n = 2 \) to the left to rewrite \( 120.054 \) in decimal form as \( 1.20054 \times 10^2 \)
    Hence \( \quad 120.054 \times 10^{-6} = 1.20054 \times 10^2 \times 10^{-6} = 1.20054 \times 10^{-4} \)

  10.  Answer B
    Solution
    Write in decimal form: \( \quad 5.9 \times 10^{-6} - 2.0 \times 10^{-7} = 0.0000059 - 0.0000002 = 0.0000057 \)
    In scientific notation: \( \quad 0.0000057 = 5.7 \times 10^{-6} \)
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