91Ó°ÊÓ

The power rule is essential when simplifying expressions involving exponents. Simply put, this rule states that when you raise a power to another power, you multiply the exponents. This applies to any terms within parentheses. For example, in the expression \left(2xy^4\right)^3, each part inside the parentheses is raised to the power of 3: 2^3, x^3, and (y^4)^3.
Breaking it down, you get:
\[2^3 = 8\]
\[x^3 = x^3\]
\[(y^4)^3 = y^{4 * 3} = y^{12}\]
This simplifies the expression inside the parentheses to 8x^3y^{12}. Remember: the power rule helps reduce complex expressions to simpler, individual terms that are easily multiplied later.

When dealing with exponents while multiplying, you add the exponents of like bases. For instance, consider two terms: 8x^3y^{12} and 4x^2y^3. To multiply these, break it down into two steps:
1. **Multiply the coefficients first**:
\(8 \times 4 = 32\)
2. **Multiply the variables by adding their exponents**:
\(x^3 \times x^2 = x^{3+2} = x^5\)
\(y^{12} \times y^3 = y^{12+3} = y^{15}\)
This way, you combine similar variables and their exponents to simplify the overall expression. Multiplication of exponents is key when working with terms involving powers, as it ensures the expression is correctly simplified.

Combining like terms can streamline an expression. In our example, as we multiply terms, we only combine those with the same base (like x and y). The multiplication process involved us combining like terms like this:
From the expression \(8x^3y^{12} \times 4x^2y^3\) to:
\(32x^5y^{15}\).
Here's the breakdown:
- **Coefficients 8 and 4 combine to make 32.**
- **Exponents of x combine: x^3 and x^2 become x^5** because you add exponents when multiplying.
- **Exponents of y combine: y^{12} and y^3 become y^{15}.**
Remember, streamlining an expression involves making sure that coefficients are combined, and like terms are neatly simplified to provide a cleaner, simplified term. Understand each step and simplify them accordingly.