Question

Multiple Choice Express \(0.0025\) in scientific notation. (Algebra Review) (A) \(2.5 \times 10^{3}\) (B) \(2.5 \times 10^{4}\) (C) \(2.5 \times 10^{-3}\) (D) \(2.5 \times 10^{-4}\)

Short answer

(C) \(2.5 \times 10^{-3}\)

Step-by-step solution

Identify the significant figures

The number 0.0025 has the significant figures 2 and 5. In scientific notation, we want to express the number as a product of a number between 1 and 10 and a power of ten. This means we need to rewrite 0.0025 as a decimal with one non-zero digit before the decimal point.

Determine the coefficient

Rewrite 0.0025 to have one non-zero digit before the decimal followed by any other significant digits. This gives us 2.5. This is our coefficient.

Count the decimal places moved

Next, count how many places the decimal point has been moved to the right to turn 0.0025 into 2.5. The decimal moves 3 places to the right: 0.0025 → 0.025 → 0.25 → 2.5.

Write as a power of ten

Since the decimal point was moved 3 places to the right, the power of ten will be negative. Therefore, 0.0025 becomes 2.5 times 10 raised to the power of negative 3. The correct expression in scientific notation is thus \(2.5 \times 10^{-3}\).

Match with options

Compare our result with the multiple-choice options: (A) \(2.5 \times 10^{3}\), (B) \(2.5 \times 10^{4}\), (C) \(2.5 \times 10^{-3}\), (D) \(2.5 \times 10^{-4}\). The correct choice is (C) \(2.5 \times 10^{-3}\).

Key concepts

Understanding Significant Figures

Significant figures are crucial in scientific notation and mathematical calculations. They are the digits in a number that contribute meaningfully to its precision. When handling significant figures, it's important to identify all non-zero digits, any zeros between significant figures, and trailing zeros in a decimal portion. In the number 0.0025, only the digits '2' and '5' are considered significant. Leading zeros, like those before the '2', are not significant because they merely indicate the decimal's position and do not contribute to the measurement's accuracy.

• Recognize that the more significant figures a number has, the more precise it is.
• Always consider significant figures when converting a number into scientific notation.

Understanding these figures ensures that you maintain measurement accuracy and consistency when rewriting numbers in scientific notation.

Exploring Powers of Ten

Powers of ten are how numbers are scaled in scientific notation, which helps express very large or small numbers in a compact form. Moving the decimal point changes the power of ten required. For example, shifting the decimal three places to the right converts 0.0025 to 2.5. This movement implies multiplying by 10 raised to a negative power, specifically \(-3\). This is because negative exponents on ten signify that we are dealing with fractions or very small values.

• The rule of thumb is: move the decimal to the right, the power is negative; move it to the left, the power is positive.
• If a decimal point is moved right 3 places to form a new number, use \(10^{-3}\).

Understanding powers of ten allows you to express numbers concisely and comprehend extreme values enthusiastically.

Deciphering Decimal Places

Decimal places are the number of digits to the right of a decimal point. They play a vital role when converting numbers to scientific notation. In our example, 0.0025, the decimal violates standard form, as only one non-zero digit should come before the decimal. Therefore, it's adjusted to show 2.5.

• Manipulating decimal places changes the power of ten needed to represent the number in scientific notation.
• Each movement of the decimal corresponds directly to a change in the exponent on ten.

Understanding decimal places lets you easily identify how far to move the decimal in scientific notation conversions, ensuring accuracy in representation.