Lesson 69: Proper Form of Scientific Notation

New Concept

Welcome to Saxon Math 1! This course builds a strong foundation by connecting arithmetic with pre-algebra, mastering the essential tools for all future math.

What’s next

To begin, we'll refine our understanding of numbers by mastering the proper form for scientific notation. You'll see examples converting expressions like $25 \times 10^{-5}$.

Property

When we write a number in scientific notation, we put the decimal point to the right of the first non-zero digit. Then, we combine the powers of 10 by adding the exponents.

Examples

$4600 \times 10^5 = (4.6 \times 10^3) \times 10^5 = 4.6 \times 10^{3+5} = 4.6 \times 10^8$
$15 \times 10^5 = (1.5 \times 10^1) \times 10^5 = 1.5 \times 10^{1+5} = 1.5 \times 10^6$
$14.4 \times 10^8 = (1.44 \times 10^1) \times 10^8 = 1.44 \times 10^{1+8} = 1.44 \times 10^9$

Explanation

Think of it like dressing a number for a science fair! First, make it neat by writing it as a number between 1 and 10. Then, combine all its 'power of 10' accessories into a single exponent by adding them together. Lookin' sharp!

Property

First write the number in proper scientific notation, then combine the powers of 10 by adding the exponents. This process works the same even when some of the exponents are negative.

Examples

$25 \times 10^{-5} = (2.5 \times 10^1) \times 10^{-5} = 2.5 \times 10^{1+(-5)} = 2.5 \times 10^{-4}$
$24 \times 10^{-7} = (2.4 \times 10^1) \times 10^{-7} = 2.4 \times 10^{1+(-7)} = 2.4 \times 10^{-6}$
$12.4 \times 10^{-5} = (1.24 \times 10^1) \times 10^{-5} = 1.24 \times 10^{1+(-5)} = 1.24 \times 10^{-4}$

Explanation

Don't let a negative exponent scare you; it just means a tiny number. The rules stay the same: add the exponents together. A positive step plus a negative step might take you backwards, and that's perfectly normal in the world of exponents!

Property

To convert a decimal smaller than 1, move the decimal point to the right to create a number between 1 and 10. The new exponent will be negative, counting the places moved. Then combine the powers.

Examples

$0.25 \times 10^4 = (2.5 \times 10^{-1}) \times 10^4 = 2.5 \times 10^{-1+4} = 2.5 \times 10^3$
$0.16 \times 10^6 = (1.6 \times 10^{-1}) \times 10^6 = 1.6 \times 10^{-1+6} = 1.6 \times 10^5$
$0.75 \times 10^{-8} = (7.5 \times 10^{-1}) \times 10^{-8} = 7.5 \times 10^{-1+(-8)} = 7.5 \times 10^{-9}$

Explanation

Tiny decimals need a makeover! To fit the 'between 1 and 10' rule, boost the number by moving its decimal point to the right. The cost of this boost is a negative exponent. Then, just add the exponents together to complete the problem.