Scientific Notation With Negative Exponents
Scientific notation with negative exponents represents very small numbers by expressing them as a decimal coefficient between 1 and 10 multiplied by a negative power of 10. To write 0.000071 in scientific notation: move the decimal right until you have one non-zero digit before the decimal point (7.1), count 5 places moved, and write 7.1 times 10 to the -5. This Grade 7 math skill from Saxon Math, Course 2 extends scientific notation to small numbers and is essential for chemistry (atomic sizes), biology (cell dimensions), and physics (wavelengths of light).
Key Concepts
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!
Common Questions
How do I write a small number in scientific notation with a negative exponent?
Move the decimal point to the right until one non-zero digit is before it. Count the moves — that is the absolute value of the negative exponent. For 0.000071: move 5 places right to get 7.1 times 10 to the -5.
What does a negative exponent in scientific notation mean?
A negative exponent means the number is very small (less than 1). The exponent tells you how many places to move the decimal to the LEFT when converting back to standard form.
How do I combine two numbers in scientific notation with negative exponents?
Multiply the decimal coefficients together and add the exponents. For example, (2 times 10 to the -3) times (4 times 10 to the -2) = 8 times 10 to the -5.
What is 5.12 times 10 to the -7 in standard form?
Move the decimal 7 places to the left: 0.000000512.
When do students learn scientific notation with negative exponents?
Scientific notation for small numbers is covered in Grade 7. Saxon Math, Course 2 covers it in Chapter 9 after introducing positive-exponent scientific notation.
What are common mistakes with negative exponent scientific notation?
Students sometimes move the decimal to the right when it should go left for negative exponents. Remember: negative exponent = small number = decimal moves left.
How is scientific notation with negative exponents used in science?
An atom's radius is about 1 times 10 to the -10 meters. A red blood cell is about 8 times 10 to the -6 meters wide. Negative exponent scientific notation makes these measurements manageable.