(4a^3b^4)(5ab^2)+(a^2b^5)(-2a^2b)

Jasonbradley

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Jun 16, 2024

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Simplifying Polynomial Expressions

This article will guide you through simplifying the polynomial expression (4a^3b^4)(5ab^2)+(a^2b^5)(-2a^2b).

Understanding the Expression

The expression consists of two multiplications of monomials. Let's break it down:

• (4a^3b^4)(5ab^2): This involves multiplying two monomials with coefficients 4 and 5, and variables 'a' and 'b' with different exponents.
• (a^2b^5)(-2a^2b): This also involves multiplying two monomials with coefficients 1 (implied) and -2, and variables 'a' and 'b' with different exponents.

Applying the Rules of Exponents

To simplify the expression, we'll use the following rules of exponents:

• Product of powers: x^m * x^n = x^(m+n)
• Coefficient multiplication: Multiply the coefficients of the monomials.

Simplifying the Expression

1. Simplify (4a^3b^4)(5ab^2):

   • Multiply the coefficients: 4 * 5 = 20
   • Multiply the 'a' terms: a^3 * a = a^(3+1) = a^4
   • Multiply the 'b' terms: b^4 * b^2 = b^(4+2) = b^6
   • Combined, we get: 20a^4b^6

2. Simplify (a^2b^5)(-2a^2b):

   • Multiply the coefficients: 1 * -2 = -2
   • Multiply the 'a' terms: a^2 * a^2 = a^(2+2) = a^4
   • Multiply the 'b' terms: b^5 * b = b^(5+1) = b^6
   • Combined, we get: -2a^4b^6

3. Combine the simplified terms:

   • 20a^4b^6 + (-2a^4b^6) = 18a^4b^6

Final Result

Therefore, the simplified form of the expression (4a^3b^4)(5ab^2)+(a^2b^5)(-2a^2b) is 18a^4b^6.