Lecture Notes: Adding and Subtracting Numbers in Scientific Notation
Key Concepts
Scientific Notation: A way to express very large or very small numbers, typically in the form of ( a \times 10^n ), where ( a ) is a coefficient and ( 10^n ) is a power of ten.
Adding/Subtracting with Same Exponents: When two numbers have the same power of ten, you can directly add or subtract their coefficients.
Adjusting Exponents: If the exponents differ, convert one number so both exponents match before performing addition or subtraction.
Examples and Solutions
Example 1: Same Exponent
Problem: ( 9 \times 10^3 - 5 \times 10^3 )
Solution:
Subtract coefficients: ( 9 - 5 = 4 )
Final Answer: ( 4 \times 10^3 )
Example 2: Same Exponent
Problems:
( 7 \times 10^4 + 2 \times 10^4 )
( 5 \times 10^6 - 3 \times 10^6 )
Solutions:
( 7 + 2 = 9 ); Answer: ( 9 \times 10^4 )
( 5 - 3 = 2 ); Answer: ( 2 \times 10^6 )
Example 3: Different Exponents
Problem: ( 12 \times 10^4 - 4 \times 10^5 )
Solution:
Convert ( 12 \times 10^4 ) to ( 1.2 \times 10^5 ) by moving the decimal left.
Subtract: ( 1.2 - 4 = -2.8 )
Final Answer: ( -2.8 \times 10^5 )
Example 4: Different Exponents
Problem: ( 3.6 \times 10^5 + 2.7 \times 10^4 )
Solution:
Convert ( 2.7 \times 10^4 ) to ( 0.27 \times 10^5 ) by moving the decimal left.
Add: ( 3.6 + 0.27 = 3.87 )
Final Answer: ( 3.87 \times 10^5 )
Example 5: Different Exponents
Problem: ( 4.2 \times 10^7 + 8 \times 10^5 )
Solution:
Convert ( 8 \times 10^5 ) to ( 0.08 \times 10^7 ) by moving the decimal right.
Add: ( 4.2 + 0.08 = 4.28 )
Final Answer: ( 4.28 \times 10^7 )
Example 6: Different Exponents
Problem: ( 0.5 \times 10^7 - 9.3 \times 10^5 )
Solution:
Convert ( 9.3 \times 10^5 ) to ( 0.093 \times 10^7 ) by moving the decimal left.
Subtract: ( 0.5 - 0.093 = 0.407 )
Final Answer: ( 0.407 \times 10^7 )
General Strategy
Check Exponents: If they are the same, directly add or subtract coefficients.
Adjust Exponents: If different, convert one number so exponents match by moving the decimal point.
Perform Operation: Add or subtract the coefficients.
Re-combine: Attach the common power of ten to the resulting coefficient.