We use the exponential form to express these extremely large and small numbers in a way that is easy to read, write, and compare. For example, From end to the edge, our galaxy, the Milky Way, is \(946,000,000,000,000,000 \mathrm{~km}\) wide. The size of certain bacteria is \(0.00000000002\) meters. All these numbers can be represented with the help of Exponential form.

When a number is multiplied by itself several times, it can be written in short form. For example, the equation for \(2 \times 2 \times 2 \times 2 \times 2=2^{5}\) and the short form the equation can be written as \(2^{5}\)  which is called an exponential expression. The factor multiplied by itself, again and again, is the base, and the number of times the factor appears is the exponent. In \(2^{5}, 2\) the base and \(5\) the exponent. We read \(2^{5}\) as \(2\) to the power \(5.\) With the help of the negative and positive exponents’ concepts, let’s understand how to express the large and small numbers in standard form.

Use of Exponents to Express Small Numbers in Standard Form

1. First, count how many digits there are from the decimal point to the last digit.
2. Simply jot down the digit that comes after the zeros if there is only one. Place a multiplication sign and write the counted digits in base \(10\) with a negative sign.
3. Suppose the number has two or more non-zero digits at the end. Then, write down the digits after the first digit, followed by a decimal point and the other non-zero digits.
4. Now subtract the number of digits appearing after the decimal point from the number calculated in the first step.
5. Place a multiplication sign and use a negative sign as an exponent to write down the counted digits.

Example: Write \(0.00000006\) in standard form.

Solution: We must move the decimal places to the right when converting a very small number like \(0.00000006\) to convert the number in its conventional form. When the exponent is shifted to the right, it becomes negative.

That is, \(6 \times 10^{-8}\)

Learn About Exponents and Powers

Application of Exponents to Express Numbers in Standard Form

We write large and small numbers in standard form to make them easier to read, and when numbers are in a standard form, it’s easy to compare. Standard form refers to any number written as a decimal number between \(1.0\) and \(10.0,\) multiplied by a power of ten. 

Example \(1.36 \times 10^{23}\) and \(0.19 \times 10^{-14}\) are represented in standard form.

Standard Form of Exponents and Powers

If a number is expressed as the product of a number between \(1\) and \(10\) and the integral power of \(10,\) it is said to be in standard form. A standard form is a method of expressing a mathematical notion, such as an equation, a number, or an expression, in a way that follows a set of rules.

\(3,500,000,000\) years is the written form of \(3.5\) billion years.

As you can see, writing a large number in its numerical form, such as \(3.5\) billion, is not only ambiguous but also time-consuming, with the risk of writing a few \(0’\)s less or more. As a result, we use the standard form to represent very large or very small numbers in a simplified manner.

With the help of exponents and powers, we can express \(3,500,000,000\) years in a standard form of \(3.5 \times 10^{9}\). Decimal is placed after \(9\) digits (towards left); therefore, the power is positive.

Use of Exponents to Express Large Numbers in Standard Form

Positive exponents can be used to express large numbers in the standard form.

Consider following numbers:

\(89,000,000,000\)
\(1,459,500,000,000\)
\(750,000,000,000,000\)
\(5,978,043,000,000,000\)

We find that reading, understanding, and comparing such large numbers is difficult. Exponents are used to making enormous numbers easier to read, interpret, and compare. 

Here the given numbers can be written in standard form as follows:

\(89,000,000,000=8.9 \times 10^{10}\)
\(1,459,500,000,000=1.459 \times 10^{12}\)
\(750,000,000,000,000=7.5 \times 10^{14}\)
\(5,978,043,000,000,000=5.978043 \times 10^{15}\)

Now, isn’t it convenient to read the numbers?

Usual Form of Exponents

Scientific notation or standard form of notation refers to a number stated in the form of \(k \times 10^{n}\) Where \(k\) is a terminating decimal so that \(1≤k<10\) and \(n\) is any integer. The number is always expressed with only one digit to the left of the decimal in normal exponential notation.

If the decimal point moves to the left or the number are large, multiply the current exponent of \(10\) by the number of places the decimal point moves.

Example: Express \(3.2 \times 10^{5}\) in the usual form.

Solution: \(3.2 \times 10^{5}\) can be written in a usual form as \(3.2 \times 100,000=3,20,000\)

Subtract the number of places the decimal moves from the current exponent of ten if the decimal point shifts to the right or is small.

Example: Express \(1.7 \times 10^{-3}\) in the usual form.

Solution: \(1.7 \times 10^{-3}\) can be written in a usual form as \(1.7 \times \frac{1}{10^{3}}=\frac{17}{1000}=0.0017\)

Solved Examples

Q.1. Express the number \(2.32 \times 10^{4}\) in the usual form.
Ans: The given number is \(2.32 \times 10^{4}\)
We write the given number in usual form as \(2.32 \times 10^{4}=2.32 \times 10000=23,200\)

Q.2. The size of a red blood cell is \(0.000007\,{\rm{m}}\) and the size of a plant cell is \(0.00001275\,{\rm{m}}{\rm{.}}\) Compare these two sizes.
Ans: We have 
Size of a red blood cell \(=0.000007 \mathrm{~m}=7 \times 10^{-6} \mathrm{~m}\)
Size of a plant cell \(=0.00001275 \mathrm{~m}=1.275 \times 10^{-5} \mathrm{~m}\)
Therefore, \(\frac{\text { Size of red blood cell }}{\text { Size of plant cell }}=\frac{7 \times 10^{-6}}{1.275 \times 10^{-5}}\)
\(=\frac{7 \times 10^{-6+5}}{1.275}=\frac{7 \times 10^{-1}}{1.275}\)
\(=\frac{0.7}{1.275}=\frac{0.7}{1.3}=\frac{1}{2}\)
So, a red blood cell is approximately half of a plant cell in size.

Q.3. If the diameters of the Sun and the Earth are \(1.4 \times 10^{9}\) metres and \(1.275 \times 10^{7}\) metres respectively. Compare these two.
Ans: We have, 
\(\frac{\text { Diameter of the } \mathrm{Sun}}{\text { Diameter of the } \mathrm{Earth}}=\frac{1.4 \times 10^{9}}{1.276 \times 10^{7}}=\frac{1.4 \times 10^{2} \times 10^{7}}{1.276 \times 10^{7}}\)
\(=\frac{1.4 \times 10^{2}}{1.276} \simeq 100\)
So, the diameter of the Sun is about \(100\) times the diameter of the Earth.

Q.4. Express the number \(7.34 \times 10^{-4}\) in the usual form.
Ans: We have, \(7.34 \times 10^{-4}=\frac{734^{4}}{10^{4}}\)
\(=\frac{7.34}{10000}=0.000734\)
Hence the usual form of \(7.34 \times 10^{-4}\) is \(0.000734\)

Q.5. Express the number 15470000000 in standard form.
Ans: To express \(15470000000\) in standard form, put decimal after \(9\) digits (towards left); therefore, the power is positive.
\(15470000000=1547 \times 10000000\)
\(=1.547 \times 1000 \times 10000000\)
\(=1.547 \times 10^{3} \times 10^{7}\)
\(=1.547 \times 10^{3+7}\)
\(=1.547 \times 10^{10}\)
Hence the standard form of \(15470000000\) is \(1.547 \times 10^{10}\)

Summary

In this article, we learnt about the use of exponents to express small numbers in standard form (steps), application of exponents to express numbers in standard form, standard form of exponents and powers, use of exponents to express large numbers in standard form, the usual form of exponents, solved examples on the use of exponents to express small numbers in standard Form and FAQs on use of exponents to express small numbers in standard form.

The learning outcome of this article is standard form helps us read and compare very large and very small numbers.

Learn About Exponents and Surds

FAQs

Q.1. How do you turn a small number into standard form?
Ans:
1. First, count the number of digits from the decimal point to the last digit. 
2. Simply scribble down the digit that follows the zeros if there is only one. 
3. Put a multiplication sign in front of the counted digits and write them in base 10 with a negative sign. Assume the number ends with two or more non-zero digits. Then, write down the digits following the first digit, followed by a decimal point and the remaining non-zero digits. 
4. Subtract the number calculated in the first step by the number of digits appearing after the decimal point.
5. To write down the counted digits, use a multiplication sign and a negative sign as an exponent.

Q.2. Why do you use the standard form to express large and small numbers?
Ans: We write very large and small numbers in standard form to make them easier to read, and when numbers are in a standard form, it’s easy to compare.
Standard form refers to any number written as a decimal number between \(1.0\) and \(10.0,\) multiplied by a power of ten. Example \(1.06 \times 10^{9}\) and \(2.3 \times 10^{-11}\) are there in standard form.

Q.3. Can very small numbers be expressed in standard form using positive exponents?
Ans: No. Very small numbers cannot be expressed in standard form using positive exponents.
Negative exponents can be used to express very small values in standard form.
Example: Write \(0.000012\) in standard form.
Solution: We must move the decimal places to the right when converting a very small number like \(0.000012\) to convert the number in its conventional form. When the exponent is shifted to the right, it becomes negative.
That is, \(1.2 \times 10^{-5} .\)

Q.4. What is the difference between exponential and standard form?
Ans: Standard Form: A quantity is said to be expressed in standard form if written as the product of a power of \(10\) and a number greater than or equal to \(1\) but less than \(10.\)
Exponential Form: Exponential notation is a different way to express numbers. The form of an exponential number is \(an,\) where \(a^{n}\) is multiplied by itself n times.

Q.5. What is the standard form in math?
Ans: Maths is defined as a representation or notation of a specific element in its standard form. Whether it’s numbers, an equation, or a line, it depends on the subject. Explanation: \(Ax+By=C\) is the standard form of a straight line. A quadratic equation’s standard form is \(a x^{2}+b x+c=0 .\)

We hope this detailed article on the use of exponents to express small numbers in standard form helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!