Multiplying with Scientific Notation
Multiplying with scientific notation in Grade 8 Saxon Math Course 3 involves multiplying the decimal coefficients and using the product rule for exponents to add the powers of 10. Students practice ensuring the final answer is in proper scientific notation form with a coefficient between 1 and 10. This operation is frequently applied in science contexts where measurements involve extremely large or small values.
Key Concepts
Property To multiply numbers in scientific notation, multiply the coefficients and add the exponents of 10: $(a \times 10^m)(b \times 10^n) = (a \cdot b) \times 10^{m+n}$.
Examples $(3 \times 10^{ 4})(2 \times 10^{ 5}) = (3 \cdot 2) \times 10^{ 4+( 5)} = 6 \times 10^{ 9}$ $(1.5 \times 10^{6})(4 \times 10^{ 2}) = (1.5 \cdot 4) \times 10^{6+( 2)} = 6 \times 10^{4}$ One dollar bill weighs about $1 \times 10^{ 3}$ kilograms. The weight of 500,000 dollar bills is $(1 \times 10^{ 3})(5 \times 10^5) = 5 \times 10^2$ kilograms.
Explanation It is a two step dance! First, multiply the front numbers, called coefficients. Then, simply add the exponents of the powers of 10 together. Be careful when adding negative exponents, as you might be making a tiny number even tinier!
Common Questions
How do you multiply two numbers in scientific notation?
Multiply the two coefficients and add the two exponents of 10. Then adjust the coefficient if needed to keep it between 1 and 10.
What is the product rule for exponents?
When multiplying powers with the same base, add the exponents: a^m x a^n = a^(m+n). Applied to scientific notation: 10^m x 10^n = 10^(m+n).
How do you multiply (2.5 x 10^3) by (4 x 10^2)?
Multiply 2.5 x 4 = 10, and add exponents 3 + 2 = 5. Result: 10 x 10^5. Normalize: 1.0 x 10^6.
Can you multiply a number in scientific notation by a regular number?
Yes. Multiply the coefficient by the regular number, keep the exponent the same, and normalize if the new coefficient is not between 1 and 10.
Why is multiplying in scientific notation useful?
It allows efficient computation with very large or very small numbers by keeping track of magnitude with exponents rather than writing out all the zeros.