To distribute an exponent, you essentially raise each term inside the parentheses to the power outside the parentheses. This principle applies differently depending on whether the terms inside the parentheses are being multiplied or divided. It's crucial to understand the underlying rules of exponents to apply this correctly.
Understanding the Basics of Exponents
An exponent indicates how many times a base number is multiplied by itself. For example, in the expression xn, x is the base, and n is the exponent. This means x multiplied by itself n times.
Distributing Exponents Over Multiplication
When dealing with terms inside parentheses that are being multiplied and the entire expression is raised to a power, you distribute the exponent to each term. The rule is:
(ab)n = anbn
Here's a breakdown:
- Identify the terms inside the parentheses: In the expression (ab)n, 'a' and 'b' are the terms being multiplied.
- Apply the exponent to each term: Each term, 'a' and 'b', is raised to the power of 'n'.
Example:
(2x)3 = 23 * x3 = 8x3
Distributing Exponents Over Division
Similarly, when dealing with terms inside parentheses that are being divided and the entire expression is raised to a power, you distribute the exponent to both the numerator and the denominator. The rule is:
(a/b)n = an / bn
Here's a breakdown:
- Identify the numerator and denominator: In the expression (a/b)n, 'a' is the numerator, and 'b' is the denominator.
- Apply the exponent to both: Both the numerator 'a' and the denominator 'b' are raised to the power of 'n'.
Example:
(x/3)2 = x2 / 32 = x2 / 9
Examples and Practical Applications
Let's look at some more examples to solidify the concept:
- (3y2)4: Apply the exponent to each factor inside the parenthesis. This becomes 34 * (y2)4 = 81y8
- (4a/b3)2: Apply the exponent to the numerator and the denominator: (4a)2 / (b3)2 = (42 * a2) / b6 = 16a2 / b6
Common Mistakes to Avoid
- Forgetting to apply the exponent to all terms: Ensure that every number, variable, and their coefficients within the parentheses receives the exponent.
- Incorrectly applying the power of a power rule: When raising a power to another power, multiply the exponents, don't add them. For example, (x2)3 = x6, not x5.
- Distributing exponents over addition or subtraction: Exponents cannot be distributed over addition or subtraction. (a + b)n ≠ an + bn. You must expand (a + b)n by multiplying (a + b) by itself n times.
Summary
Distributing exponents involves applying the exponent to each factor within the parentheses if the terms are being multiplied or divided. Remember to pay close attention to the rules of exponents and avoid common mistakes.
