Scientific Notation

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Topic Related Question Bank Categories

  • Basic Conversions.

  • Calculations with Scientific Notation.

  • Comparisons and Conceptual.

  • Word Problems.

Topic Summary & Highlights
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Core Concept

In scientific notation, a number is expressed as the product of a coefficient and a power of 10. The coefficient is a number between 1 and 10, and the power of 10 indicates the scale or magnitude of the number. The power of 10 is determined by the number of places the decimal point needs to be shifted to the left or right to obtain a coefficient between 1 and 10.

Practice Tips

  • Ensure the coefficient is between 1 and 10, apply exponent rules correctly, and round results according to significant figure rules.

  • Convert numbers between standard form and scientific notation by moving the decimal point left (for large numbers) or right (for small numbers).

  • Master multiplication rules: Multiply coefficients and add exponents, e.g., (2×103)×(3×104)=6×107

  • Master division rules: Divide coefficients and subtract exponents, e.g., (6×107)÷(2×103)=3×104.

  • Practice addition and subtraction: Align exponents before adding or subtracting coefficients, e.g., rewrite (3.2×104)+(4.5×103) as 3.2 + 0.45 = 3.65×104

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Core Concept

Scientific notation is a way to express very large or very small numbers in a more compact form. It is commonly used in scientific and mathematical calculations, as well as in representing measurements and quantities in various scientific fields.

In scientific notation, a number is expressed as the product of a coefficient and a power of 10. The coefficient is a number between 1 and 10, and the power of 10 indicates the scale or magnitude of the number. The power of 10 is determined by the number of places the decimal point needs to be shifted to the left or right to obtain a coefficient between 1 and 10.

For example:

  • The number 300,000,000 can be expressed in scientific notation as 3 x 10^8. Here, the coefficient is 3 (between 1 and 10), and the power of 10 is 8, indicating that the decimal point is shifted 8 places to the right (since its positive).

  • Similarly, the number 0.000025 can be written in scientific notation as 2.5 x 10^-5. In this case, the coefficient is 2.5, and the power of 10 is -5, indicating that the decimal point is shifted 5 places to the left (since its negative).

Performing Calculations in Scientific Notation

Adding and Subtracting

For addition and subtraction, the numbers must have the same exponent.

  1. Match the Exponents: If the exponents aren't the same, adjust one or both to create matching exponents.

  2. Add or Subtract the Coefficients: Treat the coefficients like regular numbers and perform the addition or subtraction.

  3. Maintain the Exponent: The final answer keeps the same exponent as the original numbers.

Example: Add 4.2 x 10^2 and 1.35 x 10^1

  • We need the exponents to be the same. Let's convert 1.35 x 10^1 to 1.350 x 10^2.

  • Now we can add: (4.200 + 1.350) x 10^2 = 5.55 x 10^2

Multiplication and Division

Multiplication:

  1. Multiply the coefficients.

  2. Add the exponents of the two numbers.

  3. The resulting number is in scientific notation with the new coefficient and combined exponent.

Multiply 2.5 x 10^3 by 6.2 x 10^2 = (2.5 x 6.2) x (10^3 x 10^2) = 15.5 x 10^5

Division:

  1. Divide the coefficients.

  2. Subtract the exponent of the divisor (number in the denominator) from the exponent of the dividend (number in the numerator).

  3. The resulting number is in scientific notation with the new coefficient and the difference in exponents.

Example: Divide 8.4 x 10^4 by 2.1 x 10^2

  • (8.4 / 2.1) x (10^4 / 10^2) = 4 x 10^2 (We can round 3.99 to 4 here)

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