Problem 74 - Elementary Algebra

Question

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ 3 a^{2} b^{3}\left(\frac{14 a^{2} b^{5}}{2 b}\right) $$

Step-by-Step Solution

Answer

Answer: \(21a^4b^7\)

Step 1: Distribute the constants and variables

First, let's distribute the constants and variables separately. $$ 3a^2b^3\left(\frac{14a^2b^5}{2b}\right) = \left(3 \cdot \frac{14}{2}\right)(a^2 \cdot a^2b^3 \cdot b^5 \cdot \frac{1}{b}) $$

Step 2: Simplify the constants and variables

Now, we can simplify the constants and variables: $$ \left(3 \cdot \frac{14}{2}\right)(a^2 \cdot a^2b^3 \cdot b^5 \cdot \frac{1}{b}) = 21(a^2\cdot a^2 \cdot \frac{b^3\cdot b^5}{b}) $$

Step 3: Apply the product rule for the variables

Let's apply the Product Rule for the variables: $$ 21(a^2\cdot a^2 \cdot \frac{b^3\cdot b^5}{b}) = 21(a^{2+2}\cdot \frac{b^{3+5}}{b}) $$

Step 4: Simplify the variables

Now, simplify the variables: $$ 21(a^{2+2}\cdot\frac{b^{3+5}}{b}) = 21(a^4\cdot\frac{b^8}{b}) $$

Step 5: Apply the quotient rule for the variables

Lastly, let's apply the Quotient Rule for the variables: $$ 21(a^4\cdot\frac{b^8}{b}) = 21(a^4\cdot b^{8-1}) $$

Step 6: Simplify the final expression

Simplify the final expression: $$ 21(a^4\cdot b^{8-1}) = 21a^4b^7 $$ The simplified expression is \(21a^4b^7\).

Key Concepts

Product Rule of Exponents

The product rule of exponents is a fundamental rule that simplifies the multiplication of powers with the same base. The rule states that when you multiply two exponents with the same base, you can simply add the exponents together. This turns a potentially complex expression into something more digestible.

For example, consider an expression like \(x^m \cdot x^n\). Using the product rule, you can simplify this to \(x^{m+n}\). This is immensely useful when working with algebraic expressions, as it allows us to combine terms easily.

• Ensure that the bases of the exponents are identical.
• Keep the base the same and add the exponents together.
• Express the resulting exponent as a single power of the base.

In our example from the original problem, we used \(a^2 \cdot a^2\), which simplifies to \(a^{2+2} = a^4\). Applying the product rule reduces the complexity, transforming multiplication into a simple addition problem.

Quotient Rule of Exponents

The quotient rule of exponents is an essential tool for simplifying expressions, especially when dealing with divisions. It states that when dividing two exponents with the same base, you can subtract the exponent of the denominator from the exponent of the numerator.

Mathematically, the quotient rule is expressed as \(\frac{x^m}{x^n} = x^{m-n}\). This rule efficiently handles fractions where both the numerator and denominator are powers of the same base.

• Check that the bases of the exponents in the numerator and denominator match.
• Subtract the exponents, \(m - n\), keeping the base constant.
• Write the expression as a single power.

In the simplified form of our exercise, we apply the quotient rule to \(\frac{b^8}{b}\), resulting in \(b^{8-1} = b^7\). This subtraction makes handling divisions a breeze, turning complex expressions into simpler, more manageable forms.

Simplification of Algebraic Expressions

Simplification of algebraic expressions involves breaking down complex combinations of numbers and variables into their simplest form. This process makes expressions easier to work with and understand, often involving several rules, including the product and quotient rules outlined earlier.

The goal of simplification is to transform the expression so that it only contains the most essential terms and operations, openly exposing the structure of the expression.

• Use arithmetic operations to simplify numerical coefficients.
• Apply exponent rules to condense and combine like terms.
• Seek to reduce fractions and eliminate unnecessary components.

In our problem, we began by distributing and simplifying constants like \(3 \cdot \frac{14}{2} = 21\) and then applied the rules of exponents to handle the algebraic terms. By systematically applying these techniques, we arrived at the simplified expression \(21a^4b^7\), which is much easier to interpret and utilize in further calculations.