2.05 The laws of exponents and scientific notation
We've previously looked at how to use scientific notation to write really big or really small numbers. Remember they are written in the form $a\times10^b$a×10b, where $1\le a<10$1≤a<10 and $b$b is an integer. Since these numbers are all written in relation to a power of $10$10, we can simplify expressions written in scientific notation using the laws of exponents, such as the product of powers property or the quotient of powers property.
Adding and subtracting using scientific notation
If you are adding or subtracting numbers written in scientific notation, you need to make sure that the powers of ten are the same. To do this, you may need to factor some of the powers of $10$10 based on the product of powers property. Then you can use the decimal number, rather than the number in scientific notation and evaluate the problem. The process goes:
| $a\times10^n\pm b\times10^m$a×10n±b×10m | $=$= | $a\times10^n\pm b\times10^{m-n}\times10^n$a×10n±b×10m−n×10n |
| $=$= | $\left(a\pm b\times10^{m-n}\right)\times10^n$(a±b×10m−n)×10n |
Multiplying and dividing using scientific notation
If you are multiplying or dividing numbers written in scientific notation, we will use the commutative property to multiply or divide the powers of $10$10 by one another and then the "$a$a" values by one another.
| $\left(a\times10^m\right)\left(b\times10^n\right)$(a×10m)(b×10n) | $=$= | $\left(a\times b\right)\left(10^m\times10^n\right)$(a×b)(10m×10n) |
| $=$= | $\left(a\times b\right)\times10^{m+n}$(a×b)×10m+n |
| $\frac{a\times10^m}{b\times10^n}$a×10mb×10n | $=$= | $\frac{a}{b}\times\frac{10^m}{10^n}$ab×10m10n |
| $=$= | $\frac{a}{b}\times10^{m-n}$ab×10m−n |
Powers of numbers in scientific notation
Recall the power of a power and power of a product properties. We will use both properties to take a raise a number in scientific notation to a power. The process goes:
| $\left(a\times10^m\right)^n$(a×10m)n | $=$= | $a^n\times\left(10^m\right)^n$an×(10m)n |
| $=$= | $a^n\times10^{mn}$an×10mn |
Make sure you check that your final answer is expressed appropriately in scientific notation. Simplify as required.
Worked example
Question 1
Use the laws of exponents to simplify $\left(2\times10^6\right)\times\left(6\times10^5\right)$(2×106)×(6×105). Give your answer in scientific notation.
Think: Let's simplify the expression first. Remember to express our answer in scientific notation, we'll need to express the coefficient as a value between $1$1 and $10$10 and then multiply it by the correct power of $10$10.
Do:
| $2\times10^6\times6\times10^5$2×106×6×105 | $=$= | $\left(2\times6\right)\times\left(10^6\times10^5\right)$(2×6)×(106×105) |
| $=$= | $12\times10^{6+5}$12×106+5 | |
| $=$= | $12\times10^{11}$12×1011 |
$12$12 can be as expressed as $1.2\times10^1$1.2×101. We will use this to write our answer in scientific notation.
| $1.2\times10^1\times10^{11}$1.2×101×1011 | $=$= | $1.2\times10^{1+11}$1.2×101+11 |
| $=$= | $1.2\times10^{12}$1.2×1012 |
Reflect: How might you summarize the steps above to a classmate?
Practice questions
Question 2
Evaluate $\frac{3\cdot10^3}{12\cdot10^{-1}}$3·10312·10−1, writing your answer using scientific notation.
Question 3
Evaluate $8.59\cdot10^4+1.77\cdot10^5$8.59·104+1.77·105. Give your answer using scientific notation.