Simplify (3ab)(2a^3b)
This problem is asking you to perform the operation of multiplication on two algebraic expressions. Specifically, you are to multiply the monomial 3ab with the monomial 2a^3b. Each expression consists of a numerical coefficient and variables with their respective exponents. In simplifying the product, you would apply the Commutative and Associative Properties of Multiplication, as well as the laws of exponents for multiplying like bases.
$\left(\right. 3 a b \left.\right) \left(\right. 2 a^{3} b \left.\right)$
Answer
Solution:
Process to Simplify the Expression (3ab)(2a^3b)
Step 1: Combine like terms by adding exponents of 'a'.
Step 1.1: Rearrange terms for clarity. $3a^3a \cdot b \cdot 2b$
Step 1.2: Perform the multiplication of 'a' terms.
- Step 1.2.1: Express 'a' with an exponent. $3a^3a^1 \cdot b \cdot 2b$
- Step 1.2.2: Apply the exponent rule for multiplication, $a^m \cdot a^n = a^{m+n}$. $3a^{3+1}b \cdot 2b$
Step 1.3: Calculate the sum of the exponents for 'a'. $3a^4b \cdot 2b$
Step 2: Combine like terms by adding exponents of 'b'.
- Step 2.1: Rearrange terms for clarity. $3a^4 \cdot bb \cdot 2$
- Step 2.2: Perform the multiplication of 'b' terms. $3a^4b^2 \cdot 2$
Step 3: Multiply the numerical coefficients.
- Multiply 2 by 3 to get the final coefficient. $6a^4b^2$
Knowledge Notes:
To simplify an algebraic expression involving exponents, we use several exponent rules:
Product of Powers Rule: When multiplying two expressions that have the same base, you add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.
Power of a Power Rule: When raising an exponent to another exponent, you multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.
Multiplication of Coefficients: When you multiply terms with coefficients, you multiply the coefficients together and keep the variable part the same. For example, $c \cdot a^m \cdot d \cdot a^n = cd \cdot a^{m+n}$.
In the given problem, we used the Product of Powers Rule to combine the 'a' and 'b' terms, and then we multiplied the coefficients together to simplify the expression.
