(4xy%5E3)%2B(3xy%5E4)(5y).jpeg "(2y^2)(4xy^3)+(3xy^4)(5y)")
Simplifying the Expression: (2y^2)(4xy^3)+(3xy^4)(5y)
This article explores the simplification of the given expression: (2y^2)(4xy^3)+(3xy^4)(5y). We will utilize the rules of exponents and algebraic manipulation to reach a simplified form.
Understanding the Terms
Let's break down the expression into its individual terms:
- (2y^2)(4xy^3): This term involves multiplying a constant (2) by a variable (y) raised to the power of 2, and then multiplying this by another term that includes a constant (4), a variable (x), and another variable (y) raised to the power of 3.
- (3xy^4)(5y): This term involves multiplying a constant (3) by a variable (x), another variable (y) raised to the power of 4, and finally multiplying this by a constant (5) and a variable (y).
Simplifying Each Term
To simplify each term, we apply the following rules of exponents:
- Product Rule: When multiplying exponents with the same base, you add the powers: x^m * x^n = x^(m+n)
Let's simplify each term:
-
(2y^2)(4xy^3)
- Multiply the constants: 2 * 4 = 8
- Combine the 'y' terms: y^2 * y^3 = y^(2+3) = y^5
- The simplified term becomes: 8xy^5
-
(3xy^4)(5y)
- Multiply the constants: 3 * 5 = 15
- Combine the 'y' terms: y^4 * y = y^(4+1) = y^5
- The simplified term becomes: 15xy^5
Combining the Simplified Terms
Now that we have simplified each term, we combine them:
(2y^2)(4xy^3)+(3xy^4)(5y) = 8xy^5 + 15xy^5
Finally, we combine the like terms:
8xy^5 + 15xy^5 = 23xy^5
Conclusion
Therefore, the simplified form of the expression (2y^2)(4xy^3)+(3xy^4)(5y) is 23xy^5. This process demonstrates the application of the rules of exponents and algebraic manipulation to simplify complex expressions.