Exponential Notation

Exponential notation is a way of representing a number as a product of a base and an exponent. The exponent tells us how many times the base is multiplied by itself. For example, in the expression 5^3, 5 is the base and 3 is the exponent. This means 5 is multiplied by itself 3 times: 5 * 5 * 5 = 125.

Rules for Exponential Notation

1. When multiplying two numbers with the same base, add the exponents: a^m * a^n = a^(m + n)
2. When dividing two numbers with the same base, subtract the exponents: a^m / a^n = a^(m - n)
3. When raising a power to another power, multiply the exponents: (a^m)^n = a^(m * n)

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a more compact form. It is written in the form a x 10^n, where 1 ≤ a < 10 and n is an integer. For example, the speed of light is approximately 299,792,458 meters per second. In scientific notation, this is written as 2.99792458 x 10^8.

Converting to Scientific Notation

To convert a number to scientific notation, follow these steps:

1. Move the decimal point to the right or left to create a number between 1 and 10
2. Count the number of places the decimal point was moved. This number becomes the exponent of 10

Converting from Scientific Notation

To convert a number from scientific notation to standard form, simply multiply the coefficient by 10 raised to the power of the exponent.

Study Guide

Here are some practice problems to help you master exponential and scientific notation:

1. Calculate the following using exponential notation: 2^4, 3^2, 5^3
2. Simplify the following using exponential notation: 2^3 * 2^5, 4^7 / 4^4, (3^2)^4
3. Express the following numbers in scientific notation: 480,000,000, 0.0000075, 9,600,000,000,000
4. Convert the following numbers from scientific notation to standard form: 6.4 x 10^3, 9.75 x 10^-5, 2.1 x 10^6

Practice these problems to improve your understanding of exponential and scientific notation.