Exponents are a mathematical notation that represents repeated multiplication. When we write 2 to the power of 3, we mean 2 multiplied by itself 3 times, which equals 8. The base is the number being multiplied, and the exponent tells us how many times to multiply it. Let's look at some examples: 5 squared equals 25, and 3 to the fourth power equals 81.
The three basic exponent rules form the foundation of working with exponential expressions. The product rule states that when multiplying powers with the same base, we add the exponents. For example, 2 cubed times 2 squared equals 2 to the fifth power, which is 32. The quotient rule tells us that when dividing powers with the same base, we subtract the exponents. The power rule shows that when raising a power to another power, we multiply the exponents together.
Zero and negative exponents follow logical patterns from the quotient rule. When we divide a cubed by a cubed, we get a to the zero power, which equals 1. This gives us the zero exponent rule: any non-zero number to the power of zero equals 1. Negative exponents represent reciprocals. For example, 2 to the negative 3 equals 1 over 2 cubed, which is 1 eighth. This pattern shows how exponents create a continuous sequence from positive through zero to negative values.
Fractional exponents extend our understanding to include roots. The exponent one-half represents the square root, while one-third represents the cube root. In general, a to the power of one over n equals the nth root of a. For more complex fractional exponents like m over n, we can interpret this as either the nth root of a to the m power, or the nth root of a, all raised to the m power. For example, 8 to the two-thirds power equals 4, and 16 to the three-fourths power equals 8.
Working with complex exponential expressions requires systematic application of all the rules we've learned. Let's solve three examples step by step. First, we simplify 2x cubed y squared to the fourth power divided by 4x squared y squared. Using the power rule and quotient rule, this simplifies to x to the eighth y to the sixth. Second, we have a product involving negative exponents, which simplifies to 9 a b to the fifth. Finally, we work with fractional exponents, where 8x to the sixth to the one-third power times x to the negative one equals 2x. The key is identifying which rules apply and working systematically through each step.