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Simplifying Algebraic Expressions: (3a)^2(7b)^4
In mathematics, simplifying expressions is a crucial skill that allows us to express complex formulas in a more manageable form. One common type of simplification involves expressions with exponents and variables. Let's explore how to simplify the expression (3a)^2(7b)^4.
Understanding the Rules of Exponents
To tackle this simplification, we need to recall some fundamental rules of exponents:
- Product of powers: When multiplying powers with the same base, we add the exponents: x^m * x^n = x^(m+n).
- Power of a product: When raising a product to a power, we raise each factor to that power: (xy)^n = x^n * y^n.
- Power of a power: When raising a power to another power, we multiply the exponents: (x^m)^n = x^(m*n).
Applying the Rules to Simplify
Let's apply these rules to our expression:
- (3a)^2: Using the power of a product rule, we get 3^2 * a^2 = 9a^2.
- (7b)^4: Similarly, we get 7^4 * b^4 = 2401b^4.
- (3a)^2 * (7b)^4: Now, we multiply the simplified terms: 9a^2 * 2401b^4 = 21609a^2b^4.
Conclusion
Therefore, the simplified form of (3a)^2(7b)^4 is 21609a^2b^4. By applying the fundamental rules of exponents, we can efficiently simplify complex algebraic expressions.