Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ \left(3 u^{7} v^{3}\right)\left(4 u^{4} v^{-5}\right) $$

Step-by-Step Solution

Answer

Simplified expression: \(12u^{11}v^{-2}\).

Step 1: Multiply the Coefficients

The given expression is \( (3u^7v^3)(4u^4v^{-5}) \). First, multiply the coefficients 3 and 4. This results in 12.

Step 2: Apply the Product of Powers Rule for \(u\)

Apply the product of powers rule \((a^m)(a^n) = a^{m+n}\) to the bases involving \(u\). For \(u\), you have \(u^7\) and \(u^4\). Add the exponents: \(7 + 4 = 11\). This gives you \(u^{11}\).

Step 3: Apply the Product of Powers Rule for \(v\)

Do the same for the \(v\) terms. You have \(v^3\) and \(v^{-5}\). Add the exponents: \(3 + (-5) = -2\). This results in \(v^{-2}\).

Step 4: Combine the Results

Combine the results from the previous steps. The simplified expression is \(12u^{11}v^{-2}\).

Key Concepts

Understanding the Product of Powers Rule

When dealing with algebraic expressions, the product of powers rule is a critical concept to grasp. This rule applies when multiplying two powers with the same base. According to this rule, you simply add the exponents.
For instance, if you have two terms, like \((a^m)\) and \((a^n)\), the product of these powers can be expressed as \(a^{m+n}\). This is because multiplying powers with the same base doesn’t change the base value itself, it only results in combining their exponents.
In our example, we apply the rule to the variables in our expression: both the \(u\) and the \(v\) terms.

• For \(u\), the terms are \(u^7\) and \(u^4\). Using the rule, we calculate \(u^{7+4} = u^{11}\).
• Similarly, for \(v\), we deal with \(v^3\) and \(v^{-5}\). Applying the rule here, \(v^{3 + (-5)} = v^{-2}\).

Role of Coefficients in Multiplication

Coefficients in algebraic expressions are the numerical part of the terms, found in front of the variable. When multiplying expressions, it is important to handle coefficients properly.
In our expression, the coefficients are 3 and 4. Unlike variables, coefficients do not require any rule or identification of common bases when multiplying—they merely get multiplied

• For example, multiplying 3 by 4 in our expression gives us 12.

Once the coefficients are multiplied, they are simply attached to the remaining expression from the application of other rules, such as the product of powers.

Exploring Exponents

Exponents tell us how many times a number, known as the base, is used in a multiplication. Managing exponents properly is crucial for simplifying expressions in algebra.

• For example, the term \(u^7\) in our expression indicates that \(u\) is multiplied by itself 7 times.
• Negative exponents, like \(v^{-5}\), mean you are dealing with the reciprocal of the base. For \(v^{-5}\), it's the same as \(\frac{1}{v^5}\).

Familiarity with how exponents operate, including understanding negative, zero, and fractional exponents, is key to working with and simplifying complex algebraic expressions.

Simplifying Expressions

The ultimate goal of understanding algebraic principles is to simplify complex expressions into more manageable forms. Simplifying expressions involves applying rules like the product of powers and multiplying coefficients.
In this context, our original expression \((3u^7v^3)(4u^4v^{-5})\) becomes much easier to interpret and use once simplified.

• After calculating both the coefficients and the exponents, we arrived at \(12u^{11}v^{-2}\).
• The simplified expression is not only easier to work with, but it also conveys the same mathematical value, confirming the power of simplification in algebra.

Simplification helps in making the expressions ready for further algebraic operations or evaluations that students might encounter in complex problem-solving scenarios.