%5E3.jpeg "(2x^4y^5)^3")
Simplifying (2x^4y^5)^3
In mathematics, simplifying expressions is a crucial skill. Let's tackle the simplification of (2x^4y^5)^3.
Applying the Power of a Product Rule
The key to simplifying this expression lies in understanding the power of a product rule. This rule states that when raising a product to a power, we raise each factor in the product to that power. Mathematically:
(ab)^n = a^n * b^n
Applying this to our expression:
(2x^4y^5)^3 = 2^3 * (x^4)^3 * (y^5)^3
Applying the Power of a Power Rule
Next, we encounter the power of a power rule. This rule states that when raising a power to another power, we multiply the exponents. Mathematically:
(a^m)^n = a^(m*n)
Applying this to our expression:
2^3 * (x^4)^3 * (y^5)^3 = 2^3 * x^(43) * y^(53)
Simplifying the Expression
Finally, we simplify the expression:
2^3 * x^(43) * y^(53) = 8x^12y^15
Conclusion
Therefore, the simplified form of (2x^4y^5)^3 is 8x^12y^15. This process demonstrates the importance of understanding and applying fundamental exponent rules in simplifying complex mathematical expressions.