Question

Express these numbers in scientific notation. a) 0.00656 b) 65,600 c) 4,567,000 d) 0.000005507

Short answer

a) \(6.56 \times 10^{-3}\), b) \(6.56 \times 10^{4}\), c) \(4.567 \times 10^{6}\), d) \(5.507 \times 10^{-6}\).

Step-by-step solution

Identify the Decimal Placement

For each number, decide where the decimal should be placed to have one non-zero digit to the left. This is the first step in expressing any number in scientific notation.

Calculate the Exponent for Part (a)

For 0.00656, move the decimal point 3 places to the right to get 6.56. Since the decimal is moved to the right, the exponent is negative. The scientific notation is therefore \(6.56 \times 10^{-3}\).

Calculate the Exponent for Part (b)

For 65,600, move the decimal point 4 places to the left to get 6.56. Since the decimal is moved to the left, the exponent is positive. The scientific notation is therefore \(6.56 \times 10^{4}\).

Calculate the Exponent for Part (c)

For 4,567,000, move the decimal point 6 places to the left to get 4.567. Since the decimal is moved to the left, the exponent is positive. The scientific notation is therefore \(4.567 \times 10^{6}\).

Calculate the Exponent for Part (d)

For 0.000005507, move the decimal point 6 places to the right to get 5.507. Since the decimal is moved to the right, the exponent is negative. The scientific notation is therefore \(5.507 \times 10^{-6}\).

Key concepts

Decimal Placement

Decimal placement involves determining where to place the decimal point in a number so that there is just one non-zero digit to the left of the decimal. This is the first step in converting a number to scientific notation. It's essential because scientific notation requires the number to be expressed in the form of \[a \times 10^n\], where 1 ≤ \(a\) < 10.
For example:

• In the number 0.00656, move the decimal point three places to the right to make it 6.56. Thus, the decimal placement involves moving the decimal till you reach 6.56.
• In contrast, for 65,600, move the decimal four places to the left to get 6.56.

Remember, correctly placing the decimal point sets the stage for calculating the appropriate exponent, whether negative or positive.

Significant Figures

Significant figures are the digits in a number that contribute to its precision. When expressing numbers in scientific notation, the significant figures are held constant, altering only the placement of the decimal. This ensures you retain the original accuracy of the number while adjusting its scale.
For example:

• In the number 0.00656, the significant figures are 6, 5, and 6, so in scientific notation, you represent it as 6.56.
• Similarly, for the number 4,567,000, the significant figures are 4, 5, 6, and 7, which are kept in the scientific notation form as 4.567.

Understanding and identifying significant figures ensures you don't lose precision when converting a number into scientific notation.

Exponents Calculation

After determining the decimal placement and identifying the significant figures, the last step in scientific notation is calculating the exponent. This exponent signifies the number of decimal places the point has shifted. The direction of this shift determines if the exponent is positive or negative.
Here's how it works:

• For numbers greater than 1, like 65,600, the decimal moves left, resulting in a positive exponent. Four shifts left yield an exponent of +4, so 65,600 becomes \(6.56 \times 10^4\).
• For numbers less than 1, such as 0.000005507, the decimal moves right, leading to a negative exponent. Six moves to the right make the exponent -6, represented as \(5.507 \times 10^{-6}\).

These exponents reflect the number's size and scale, streamlining and optimizing how we handle large or small values in scientific notation.