Problem 53 - Beginning and Intermediate Algebra

Question

Convert to scientific notation. $$0.000000437$$

Step-by-Step Solution

Answer

The short answer is: \(4.37 × 10^{-7}\).

Step 1: Identify the Number

Convert to scientific notation.
$$0.000000437$$

Step 2: Move the Decimal

Write as \(a \times 10^n\) where \(1 \le |a| < 10\).

Step 3: Result

The short answer is: \(4.37 × 10^{-7}\).

Key Concepts

Decimal Point

The decimal point is a crucial part of understanding decimal numbers and scientific notation. In decimal notation, it separates the integer part of a number from the fractional part. For example, in the number 0.000000437, the decimal point separates the leading zeros from the non-zero digits.

When converting to scientific notation, adjusting the decimal point is vital for creating a manageable number. We want a number, known as 'a', that lies between 1 and 10. To achieve this, you adjust the position of the decimal point.

In 0.000000437, moving the decimal point 7 places to the right yields 4.37, as required for scientific notation.

Powers of 10

Powers of 10 are essential when converting numbers into scientific notation. These powers allow us to express very large or very small numbers succinctly using exponents. In this context, a power of 10 indicates how many times you multiply the base number, 10, by itself.

Positive exponents represent large numbers, while negative exponents denote small numbers. For 0.000000437, moving the decimal point to form the number 4.37 required shifting it 7 places to the right.

This movement is represented by the power \(10^{-7}\), indicating that 4.37 is multiplied by \(10^{-7}\), to accurately express the original number.

Conversion Process

The conversion process to scientific notation involves a few straightforward steps. First, identify a number between 1 and 10, which will be the result of adjusting the decimal point position. In the example 0.000000437, moving the decimal point 7 places to the right gives you 4.37.

Next, determine the power of 10. This power is equal to the number of places you moved the decimal point. Moving it to the right means you'll have a negative exponent. Here, moving 7 places gives \(n = -7\).

Finally, combine these elements into the scientific notation format, \(a \times 10^n\). So, 0.000000437 becomes \(4.37 \times 10^{-7}\), compactly representing your original number.