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

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.

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: $5.1 \times 10^{5}$

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 $100000 + 3,000000 = 3.1 \times 10^{6}$

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: $2.8 \times 10^{-6}$

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

Solution
We are given the scientific notation: $2.3 \times 10^{-5} = \dfrac{2.3}{100000} = 0.000023$

Solution
Write in decimal form: $\quad \dfrac{1}{10000}+\dfrac{2}{1000000} = 0.0001 + 0.000002 = 0.000102$

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

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: $\quad 2.1 \times 10^{-5} + 3.0 \times 10^{-4} = 0.000021 + 0.0003 = 0.000321 = 3.21 \times 10^{-4}$

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 $\quad 120.054 \times 10^{-6} = 1.20054 \times 10^2 \times 10^{-6} = 1.20054 \times 10^{-4}$
Write in decimal form: $\quad 5.9 \times 10^{-6} - 2.0 \times 10^{-7} = 0.0000059 - 0.0000002 = 0.0000057$
In scientific notation: $\quad 0.0000057 = 5.7 \times 10^{-6}$