To simplify brackets with powers, you need to understand and apply the rules of exponents. Here's a breakdown of how to do it:
Understanding the Basics
When you have an expression inside brackets raised to a power, you're essentially multiplying the expression by itself a certain number of times (the power). The key is to distribute the power to each term inside the bracket if it's a product or a quotient.
Rules to Apply
Here's a table summarizing the important rules:
| Rule | Example | Explanation |
|---|---|---|
| Power of a Product: (ab)^n = a^n b^n | (2x)^3 = 2^3 * x^3 = 8x^3 | Distribute the power to each factor inside the brackets. |
| Power of a Power: (a^m)^n = a^(m*n) | (x^4)^3 = x^(4*3) = x^12 | Multiply the exponents when raising a power to another power. |
| Power of a Quotient: (a/b)^n = a^n/b^n | (x/y)^2 = x^2/y^2 | Distribute the power to both the numerator and the denominator. |
Steps to Simplify
- Identify the Expression: Determine what is inside the brackets and what the power is outside the brackets.
- Apply the Power of a Product/Quotient Rule: Distribute the outer power to each term inside the brackets.
- Apply the Power of a Power Rule: If any term inside the brackets already has a power, multiply that power by the power outside the brackets.
- Simplify Numerical Coefficients: Evaluate any numerical powers (e.g., 2^3 = 8).
- Combine Like Terms (if possible): If there are any like terms after applying the exponent rules, combine them to further simplify.
Examples
-
(2x^2y)^3
- Apply power of a product: 2^3 * (x^2)^3 * y^3
- Simplify: 8x^6y^3
-
(yx^4)^3 from the reference.
- Apply power of a product: y^3 * (x^4)^3
- Apply power of a power: y^3 * x^(4*3)
- Simplify: y^3x^12
-
((a/b)^2)^3
- Apply power of a power: (a/b)^(2*3) = (a/b)^6
- Apply power of a quotient: a^6 / b^6
Important Notes
- The power only applies to what is inside the brackets directly preceding it.
- Be careful with negative signs. If a negative number is raised to an even power, the result is positive. If raised to an odd power, the result is negative. Example: (-2x)^2 = 4x^2, but (-2x)^3 = -8x^3.
By following these rules and steps, you can effectively simplify expressions involving brackets with powers.
