Question

Write the following numbers in powers of 10 notation: (a) 1.156, (b) 21.8, (c) 0.0068, (d) 328.65, (e) 0.219, and (f) 444.

Short answer

The numbers in powers of 10 notations are (a) \(1.156 \times {10^0}\), (b) \(2.18 \times {10^1}\), (c) \(6.8 \times {10^{ - 3}}\), (d) \(3.2865 \times {10^2}\), (e) \(2.19 \times {10^{ - 1}}\), and (f) \(4.44 \times {10^2}\).

Step-by-step solution

Meaning of scientific notation

Scientific notation is the method of writing any number in the form of powers of 10. It is a convenient method for expressing very large or small numbers. It is written in the form of decimal numbers along with powers of 10.

Steps to write the powers of 10 notation

• Place the decimal just after the first digit of the number.
• Next, add the multiplication operator and write 10 to the power of the number of times ten repeats itself. It represents the number of places to move forward from the decimal point.

For example, the number 5000 can be represented as \(5 \times {10^3}\) in the powers of 10 notation.

(a) Conversion of the number 1.156 in the powers of 10 notation

In the given number, a decimal is already placed after the first digit. So, the power of 10 will be zero.

Thenumber1.156 can beexpressed in thepowers of 10 notation as

\(1.156 = 1.156 \times {10^0}.\)

(b) Conversion of the number 21.8 in the powers of 10 notation

Multiply and divide the given number by 10. Then, put a decimal after the digit 2 and add 1 to the power of 10.

Thus, the number 21.8 can be expressed inthepowers of 10 notation as follows:

\(\begin{aligned}{c}21.8 = 21.8 \times \frac{{10}}{{10}}\\ = \frac{{21.8}}{{10}} \times 10\\ = 2.18 \times {10^1}\end{aligned}\)

(c) Conversion of the number 0.0068 in the powers of 10 notation

In the given number, there are two zeros before the non-zero digit (6). Add a decimal after the digit 6 to get three digits before the decimal. Now, add \(\left( { - 3} \right)\) to the power of 10.

Thus, thenumber0.0068 can beexpressed in thepowers of 10 notation as follows:

\(\begin{aligned}{c}0.0068 = 0.0068 \times \frac{{1000}}{{1000}}\\ = \left( {0.0068 \times 1000} \right) \times \frac{1}{{1000}}\\ = 6.8 \times {10^{ - 3}}\end{aligned}\)

(d) Conversion of the number 328.65 in the powers of 10 notation 

Multiply the numerator and the denominator by 100. Then, put a decimal after the digit 3 and add 2 to the power of 10.

Thus, thenumber328.65 can beexpressed in thepowers of 10 notation as follows:

\(\begin{aligned}{c}328.65 = 328.65 \times \frac{{100}}{{100}}\\ = \left( {328.65 \times \frac{1}{{100}}} \right) \times 100\\ = 3.2865 \times {10^2}\end{aligned}\)

(e) Conversion of the number 0.219 in the powers of 10 notation

In this number, decimal is already there before the first digit. Multiply the numerator and the denominator by 10. Next, put a decimal after the digit 2 and add \(\left( { - 1} \right)\) to the power of 10.

Thus, thenumber0.219 can beexpressed in thepowers of 10 notation as follows:

\(\begin{aligned}{c}0.219 = 0.219 \times \frac{{10}}{{10}}\\ = \left( {0.219 \times 10} \right) \times \frac{1}{{10}}\\ = 2.19 \times {10^{ - 1}}\end{aligned}\)

(f) Conversion of the number 444 in the powers of 10 notation 

In this number, multiply the numerator and the denominator by 100. Then, put a decimal after the digit 4 and add 2 to the power of 10.

Thus, thenumber444 can beexpressed in thepowers of 10 notation as follows:

\(\begin{aligned}{c}444 = 444 \times \frac{{100}}{{100}}\\ = \frac{{444}}{{100}} \times 100\\ = 4.44 \times {10^2}\end{aligned}\)