Problem 30 - Algebra A Combined Function

Question

Use the product rule to simplify each expression. $$ \left(-7 a^{3} b^{3}\right)\left(7 a^{19} b\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-49a^{22}b^4\).

Step 1: Understand the Product Rule

The product rule for exponents states that when multiplying two expressions that have the same base, you can add the exponents. That is, for any real number \(x\) and integers \(m\) and \(n\), \(x^m \times x^n = x^{m+n}\).

Step 2: Identify Like Terms

In the expression \((-7a^3b^3) \times (7a^{19}b)\), identify the terms that share the same base. The like terms are: \(a^3\) and \(a^{19}\) as well as \(b^3\) and \(b\).

Step 3: Multiply the Coefficients

Multiply the coefficients of both terms. The coefficients are -7 and 7. So, \(-7 \times 7 = -49\).

Step 4: Apply the Product Rule to the 'a' Terms

Apply the product rule to the terms with base \(a\). Multiply \(a^3 \times a^{19}\) by adding the exponents: \(a^{3+19} = a^{22}\).

Step 5: Apply the Product Rule to the 'b' Terms

Apply the product rule to the terms with base \(b\). Multiply \(b^3 \times b\) by adding the exponents: \(b^{3+1} = b^4\).

Step 6: Combine All Parts

Combine the results of previous steps: the coefficients product and the products of like bases. The simplified expression is \(-49a^{22}b^4\).

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication of the same factor. For example, in the expression \(a^3\), the \(3\) is the exponent, indicating that \(a\) is multiplied by itself 3 times: \(a \times a \times a\). Exponents can make working with large numbers more manageable and help simplify mathematical expressions. They are core components of algebra and are crucial when applying the Product Rule.

• Base: This is the number that is being multiplied.
• Exponent: This tells you how many times to multiply the base by itself.

When using exponents, we leverage the Product Rule to simplify expressions, particularly when dealing with terms that have the same base. This rule is fundamental when tackling problems related to exponents and works by adding the exponents of terms with the same base.

Coefficient Multiplication

Coefficient multiplication is a crucial part of simplifying algebraic expressions. Coefficients are the numerical parts of terms in a mathematical expression. In the expression \(-7a^3b^3\), \(-7\) is the coefficient.When multiplying terms, start by multiplying these coefficients separately before dealing with the variables and their exponents. For example, given the terms \(-7a^3b^3\) and \(7a^{19}b\), multiply the coefficients \(-7\) and \(7\), which results in \(-49\).Remember these key points when multiplying coefficients:

• Always multiply the numerical (coefficients) part first.
• Consider the sign: multiplying positive and negative numbers can change the sign of the result.
• Combine your results with the simplified variable part to get the full simplified expression.

Simplifying Expressions

Simplifying expressions is the process of condensing a mathematical expression into its simplest form. This involves applying mathematical rules and operations to reduce complexity. In algebra, simplifying expressions often involves:

• Combining like terms.
• Applying the Product Rule by adding exponents for terms with the same base.
• Multiplying coefficients.

For instance, the expression \((-7a^3b^3)(7a^{19}b)\) can be simplified using these steps:- First, multiply the coefficients \(-7\) and \(7\) to get \(-49\).- Use the Product Rule for the \(a\) terms, \(a^3 \times a^{19} = a^{22}\).- Apply the Product Rule for the \(b\) terms, \(b^3 \times b = b^4\).- Combine these results to get the simplified expression: \(-49a^{22}b^4\).Following these procedures ensures a clear, concise, and fully simplified expression, making it easier to handle and understand.