Which number in each of the following pairs is smaller?a. 4.9 x 10⁻³ or 5.5 x 10⁻⁹

Verified step by step guidance

1. Step 1: Understand that both numbers are in scientific notation, which is a way to express very large or very small numbers.
2. Step 2: Compare the exponents of 10 in both numbers. The number with the smaller exponent is generally smaller.
3. Step 3: In this case, compare the exponents: -3 and -9. Since -9 is smaller than -3, the number with the exponent -9 is smaller.
4. Step 4: If the exponents were the same, you would then compare the coefficients (4.9 and 5.5) to determine which is smaller.
5. Step 5: Conclude that 5.5 x 10⁻⁹ is the smaller number because it has the smaller exponent.

Key Concepts

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is represented as a product of a number (between 1 and 10) and a power of ten. For example, 4.9 x 10⁻³ means 4.9 divided by 1000, which helps in comparing very small or very large values easily.

Comparing Exponents

When comparing numbers in scientific notation, the exponent of ten plays a crucial role. A larger negative exponent indicates a smaller number. For instance, 10⁻³ (0.001) is larger than 10⁻⁹ (0.000000001), meaning that any number multiplied by 10⁻³ will be larger than the same number multiplied by 10⁻⁹, regardless of the coefficients.

Decimal Conversion

To compare numbers in scientific notation directly, converting them to decimal form can be helpful. For example, 4.9 x 10⁻³ converts to 0.0049, while 5.5 x 10⁻⁹ converts to 0.0000000055. This conversion allows for straightforward comparison of the two values, making it easier to determine which is smaller.