Which number in each of the following pairs is larger? a. 7.2 x 10³ or 8.2 x 10²

Verified step by step guidance

1. Convert both numbers to the same power of ten for easy comparison.
2. 7.2 \times 10^3 is already in the form of 10^3.
3. Convert 8.2 \times 10^2 to 10^3 by multiplying by 10: 8.2 \times 10^2 = 0.82 \times 10^3.
4. Now compare 7.2 \times 10^3 and 0.82 \times 10^3.
5. Since 7.2 is greater than 0.82, 7.2 \times 10^3 is the larger number.

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 typically formatted as a number between 1 and 10 multiplied by a power of ten. For example, 7.2 x 10³ means 7.2 multiplied by 1000, which simplifies calculations and comparisons of large numbers.

Comparing Exponents

When comparing numbers in scientific notation, the exponent plays a crucial role. The number with the larger exponent is generally larger, provided the coefficients (the numbers before the exponent) are similar. In this case, comparing 10³ and 10² shows that 10³ is larger, indicating that any number multiplied by it will also be larger if the coefficients are not significantly different.

Coefficient Comparison

After establishing the exponents, the next step is to compare the coefficients of the numbers. If the exponents are the same, the larger coefficient determines the larger number. In the given pairs, if the exponents were equal, we would directly compare 7.2 and 8.2 to determine which is larger.