When multiplying variables with exponents, you first multiply the bases together and then you keep the exponents the same. This is due to the fourth exponent rule, which states that when raising several variables by a power, distribute the power to each base.
Understanding the Rules
Multiplying variables with exponents can seem complex, but it follows a few key principles. The most important thing to remember is to deal with the bases and the exponents separately.
Key Concept:
The fourth exponent rule is crucial for understanding how exponents interact when multiplying variables that are raised to a power.
Here is a detailed breakdown:
- Identify the Bases: A base is the variable or number being raised to a power.
- Identify the Exponents: An exponent indicates how many times the base is multiplied by itself.
- Multiplication: When multiplying terms with the same base, exponents are added together. However, if the bases are different, and are raised to a power, that power is distributed to each variable, and the exponents remain the same for each.
Examples
Here are some examples to illustrate the concepts of multiplying variables with exponents:
| Original Expression | Simplified Form | Explanation |
|---|---|---|
| (a * b)2 | a2b2 | The exponent 2 is distributed to both bases 'a' and 'b'. The exponents of a and b were both 1, and they remain the same after being raised to the power of 2. |
| (2xy)3 | 23x3y3 = 8x3y3 | The exponent 3 applies to all bases (2, x, and y). The exponent on each variable is 1, and remains 1 when distributed. |
Practical Insights
Here are some useful tips:
- Distribute Carefully: When variables are within parentheses with an exponent outside the parenthesis, make sure to distribute the exponent to each base, including numerical coefficients.
- Numerical Coefficients: Don't forget to apply the exponents to numerical coefficients as well.
- Simplify Further: After distributing, simplify any numerical coefficients by raising them to the indicated power.
Conclusion
In essence, when multiplying variables with exponents, first multiply the bases, and then the exponent is distributed to each variable and remains the same. Keeping the exponent distributed is critical to correctly simplifying and manipulating these expressions.
