Question

Write the number 0.000066600 in scientific notation with five significant figures.

Short answer

6.6600 x 10^{-5}

Step-by-step solution

Identify the Significant Figures

To write a number in scientific notation with significant figures, first identify the significant figures in the given number. The number 0.000066600 has five significant figures, which are 6, 6, 6, 0, and 0.

Convert to Standard Scientific Notation

Convert the number into scientific notation. The goal is to express it in the form of \(a \times 10^n\), where \(a\) is a number between 1 and 10. Move the decimal point in 0.000066600 to the right until only one non-zero digit remains to the left of the decimal point. This means moving it 5 places to the right, turning it into 6.6600.

Determine the Exponent

The decimal was moved 5 places to the right to convert 0.000066600 to the form 6.6600. Since you moved it to the right, the exponent \(n\) is negative. Thus, the exponent is -5.

Construct the Final Scientific Notation

Combine the number from Step 2 with the exponent from Step 3 to write in scientific notation. The final scientific notation for 0.000066600 with five significant figures is \(6.6600 \times 10^{-5}\).

Key concepts

Significant Figures

When dealing with significant figures, you refer to the digits that contribute to the precision of a number. In the number 0.000066600, the significant figures are not just the non-zero digits, but also include any trailing zeros that are important to indicating the precision of the number.

To identify the significant figures:

• Ignore leading zeros, as they are only placeholders and do not add precision to the number.
• Include all non-zero digits. In our example, the digits 6, 6, and 6 are significant.
• Include any trailing zeros if they are to the right of a non-zero digit and a decimal point. In 0.000066600, the zeros after the last 6 are significant because they show precision, bringing the total to five significant figures: 6, 6, 6, 0, and 0.

Paying attention to significant figures ensures that you accurately represent the number's precision when converting to scientific notation.

Decimal Point

The decimal point in a number separates the whole number from the fractional part. When converting a number to scientific notation, you move the decimal point to create a new number where there is only one non-zero digit to its left. This new number becomes the coefficient in scientific notation.

To achieve this:

• Locate the decimal point in your number. In 0.000066600, it's right at the start of the significant figures.
• Shift the decimal point to the right until only one non-zero digit is to the left of it. In our example, the decimal point is moved 5 places to the right, making the number 6.6600.
• This 6.6600 now serves as the base of our scientific notation, correctly reflecting the original number with the intended precision.

Remember, every movement of the decimal point influences the exponent of 10, indicating how many places the number has been scaled.

Exponent

The concept of an exponent in scientific notation precisely indicates how many places you move the decimal point to transform the original number into the new coefficient. This movement direction also determines the sign of the exponent.

Here's how the exponent works:

• Count the number of places you shift the decimal point from the original location to get to your new number. For 0.000066600, moving it 5 places to the right achieves this.
• If you move the decimal to the right, as we did, the exponent is negative, representing a number smaller than one. Hence, the exponent is -5, meaning our scientific notation is scaled down by five places.
• The scientific notation representation for 0.000066600 becomes 6.6600 × 10^-5, where the exponent -5 accurately reflects the decimal point movement.

Understanding the use of an exponent is key to correctly expressing numbers in scientific notation, especially when denoting very large or very small quantities.