Simplify (3ab)(2a^3b)

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. $3a^{3}a\cdot b\cdot 2b$

• Step 1.2: Perform the multiplication of 'a' terms.

  • Step 1.2.1: Express 'a' with an exponent. $3a^3{a}^1\cdot b\cdot 2b$
  • 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'. $3a^{4}b\cdot 2b$

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. $3a^4{b}^2\cdot 2$

Step 3: Multiply the numerical coefficients.

• Multiply 2 by 3 to get the final coefficient. $6a^4{b}^2$

Knowledge Notes:

To simplify an algebraic expression involving exponents, we use several exponent rules:

1. Product of Powers Rule: When multiplying two expressions that have the same base, you add the exponents. For example, $a^m\cdot a^n=a^{m+n}$.

2. Power of a Power Rule: When raising an exponent to another exponent, you multiply the exponents. For example, $(a^m{)}^n=a^{m\cdot n}$.

3. 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.