Question

SCIENTIFIC NOTATION Rewrite in scientific notation. $$ 0.00032954 $$

Short answer

The number 0.00032954 in scientific notation is \( 3.2954 \times 10^{-4} \).

Step-by-step solution

Identify the Decimal Point Position

In our number, 0.00032954, the decimal point is four places to the right from the first non-zero digit, which is 3.

Rewriting in Scientific Notation

Considering the number of movements the decimal point had to make, we can rewrite the number in the form: \( 3.2954 \times 10^{-4} \). The negative exponent is used since the number we started with is smaller than 1.

Key concepts

Decimal Point Position

Understanding the position of a decimal point is critical when it comes to scientific notation, a method used to simplify large or very small numbers. For this discussion, let's focus on numbers less than one, such as the example given: 0.00032954. Here, the decimal point is not where we want it for scientific notation. We need it to be after the first non-zero digit.

In our original number, the first non-zero digit is 3, and it is four positions to the right of the decimal point. To express this number in scientific notation, we must shift the decimal point so that it sits just to the right of the first significant figure (in this case, 3), resulting in 3.2954. Counting how many places the decimal moves gives us the exponent for the base 10 in our scientific notation. As the move is to the right for numbers smaller than one, the exponent will be negative, signifying a small number.

Rewriting Numbers

From Standard Form to Scientific Notation

When rewriting numbers in scientific notation, our goal is to make them easier to read, compare, and use in calculations. The process involves placing the number between 1 and 10 multiplied by a power of 10. For instance, the number 0.00032954 is rewritten as 3.2954 when we adjust the decimal point. The crux of rewriting a number into scientific notation lies in keeping the value of the number unchanged while adjusting its form.

Once we've identified the form as 3.2954, we combine it with the power of 10 that corresponds to the number of places the decimal point moved. Since it moved four places, we use the exponent -4 with the base 10. Thus, the number 0.00032954 is accurately represented as 3.2954 × 10^-4 in scientific notation. This method streamlines working with extremely small values, making scientific notation a staple in fields requiring precision and clarity.

Negative Exponent

The concept of a negative exponent is often confusing at first, but it is quite simple once understood. In scientific notation, a negative exponent indicates that we are dealing with a number less than one. A negative exponent means the base (in this case, 10) is to be divided into 1, rather than multiplied by itself.

So, when we say 3.2954 times 10 to the power of -4, or in mathematical terms, 3.2954 × 10^-4, what we're actually saying is 3.2954 divided by 10,000 (since 10^4 equals 10,000). Imagine 'pushing' the decimal point to the left for each count of the negative exponent, which, for our number, means we move it left four times. This gives us back our original number, 0.00032954, confirming that the scientific notation expresses the same value, just in a more compact form.