To multiply brackets involving indices, you need to apply the power to everything inside the bracket. This often involves distributing the exponent to each term and variable within the brackets.
Understanding the Basics
When a bracket with terms and indices is raised to a power, it signifies that everything inside the bracket is multiplied by itself as many times as the power indicates. For example, (a b)^n is equal to a^n b^n.
Steps to Multiply Brackets with Indices
- Identify the Bracket and Power: Look for brackets with terms inside, raised to a power.
- Distribute the Power: Apply the outer power to each term inside the bracket. Remember that each number and variable has an implicit power of 1 which is then multiplied by the outer power.
- Simplify: Multiply and simplify any terms with indices after distributing.
- Final Result: Write the final result, ensuring all powers are applied correctly.
Example
Let's take the example from the provided reference:
(3k²)³. This means (3k²) multiplied by itself three times: (3k²)(3k²)(3k²). Here's how to solve it:
- Distribute the power: (3¹ k²)^3 becomes 3¹³ k²³, which is 3³ * k⁶.
- Simplify: 3³ equals 27, so the simplified expression is 27k⁶.
Detailed Breakdown
Step-by-step Example
Consider (2xy²)³:
- Distribute the exponent: 2³ x³ (y²)³.
- Apply the power to the existing index: 2³ x³ y²*³.
- Simplify: 8 x³ y⁶.
Explanation
- Numbers: Any number within the bracket is raised to the outside power. If there is no written power on a number, it is treated as having a power of 1.
- Variables: The outside power multiplies the existing power of the variable. If no power is written, the power is 1.
- Multiple Terms: Distribute the power to every term inside the bracket, whether it’s a single term with multiple variables or multiple terms within the bracket, as in (a + b)^n.
- Remember: An expression such as (a + b)^2 is NOT equal to a^2 + b^2. (a+b)^2 = (a+b)(a+b) = a^2+2ab+b^2.
Practical Insights
- Negative Exponents: If you encounter a negative power outside of the bracket, remember that the negative power applies to all terms inside the bracket. Example (2x)^-2 = 2^-2x^-2 = 1/2^2 1/x^2 = 1/(4x^2)
- Fractional Exponents: Fractional exponents represent roots. For example, (x^4)^(1/2) = x^2.
- Order of Operations: Remember to apply the order of operations. Indices should be handled before multiplication or division.
Table of Key Concepts
| Concept | Explanation | Example |
|---|---|---|
| Power of a Product | When a product is raised to a power, each factor gets that power. | (ab)ⁿ = aⁿbⁿ |
| Power of a Power | When a power is raised to another power, multiply the exponents. | (aⁿ)ᵐ = aⁿᵐ |
| Distributing Power | Apply the exponent outside the bracket to every term inside. | (2x²y)³ = 2³x⁶y³ |
| Numbers with No Power | When a number is not written with an index, it is treated as having the power of one | 3 = 3¹ |
| Variables with No Power | When a variable is not written with an index, it is treated as having the power of one | x = x¹ |
By following these steps and understanding the underlying principles, multiplying brackets involving indices becomes a manageable process.
