Problem 47 - 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. $$ \left(\frac{1}{4} a^{2} b^{4}\right) \quad\left(\frac{1}{2} b^{4}\right) $$

Step-by-Step Solution

Answer

Question: Simplify the following expression using the product rule and quotient rule of exponents: $$\left(\frac{1}{4} a^{2} b^{4}\right) \quad\left(\frac{1}{2} b^{4}\right)$$ Answer: $$\frac{1}{8}a^2b^8$$

Step 1: Using the product rule with the coefficients

First, we need to multiply the coefficients of the given expression (1/4 and 1/2). $$ \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} $$

Step 2: Multiplying the terms with the same base

Now, multiply the terms with the same base together using the product rule of exponents, i.e., by adding exponents with the same base. There is only one \(a^2\) term, we do not need to operate on it. For the terms with the base of 'b', we have \(b^4\) and \(b^4\). Adding the exponents, we get: $$ b^{(4 + 4)} = b^8 $$

Step 3: Combining the coefficients and the terms with the base

Now we can combine the results from Step 1 and Step 2 to get the simplified expression: $$ \frac{1}{8}a^2b^8 $$ Hence, the simplified expression using the product rule and quotient rule of exponents is: $$ \frac{1}{8}a^2b^8 $$

Key Concepts

Understanding the Product Rule of Exponents

When working with exponents, one crucial concept is the **product rule**. This rule helps us simplify expressions where we are multiplying terms with the same base. The product rule states: when multiplying two powers that have the same base, you can add their exponents. This is key to simplifying expressions with exponents.
For example, let's look at something like \( b^m \times b^n \). Instead of multiplying the base \(b\) multiple times, we can simply add the exponents \(m\) and \(n\), resulting in \( b^{m+n} \). This method is a shortcut that makes calculations much easier.

• The base must be identical for the product rule to apply.
• Only add the exponents; do not multiply them.
• The order of multiplication does not change the outcome due to the commutative property of multiplication.

This rule is a powerful tool to have in your math toolkit, as it can greatly reduce the complexity of problems involving exponential terms.

Using the Quotient Rule of Exponents

The **quotient rule** comes into play when you divide expressions with the same base. It allows you to simplify such expressions by subtracting the exponents. This is particularly useful when trying to simplify complex expressions.Consider the expression \( a^m \div a^n \). Instead of calculating large powers, you can subtract the exponent \(n\) from \(m\), resulting in \( a^{m-n} \). Employing this rule reduces the expression significantly.

• The divisor's exponent is subtracted from the dividend's exponent.
• This rule only applies when the bases are the same and nonzero.
• Always ensure the subtraction results in a non-negative exponent.

Remembering these details will make working with fractional exponents much simpler, especially in algebraic contexts.

Simplifying Expressions with Exponents

Simplifying mathematical expressions is all about making them easier to understand and work with. When it comes to exponents, using rules like the product and quotient rules can turn complex-looking expressions into something much more manageable. Simplification involves applying all rules and operations correctly.In the step-by-step solution for the given exercise, you'd notice first multiplying the coefficients: \(\frac{1}{4}\) and \(\frac{1}{2}\), resulting in \(\frac{1}{8}\). After dealing with coefficients separately, the next step is to handle the variables. By applying the product rule to the terms with same bases, such as combining \( b^4 \) and \( b^4 \) into \( b^8 \), the expression is heavily simplified.

• Ensure coefficients and terms with equal bases are treated separately before combining.
• Focus on achieving the simplest form by methodically applying rules.
• Simplifying can involve multiple steps, always track your work.

This laid-back and step-by-step approach makes simplifying expressions easier to digest.

Working with Whole Numbers in Exponents

Exponents, when used with whole numbers, are straightforward and easy to calculate. A whole number exponent refers to the number of times you multiply a number by itself. For instance, in \( a^2 \), the number \(a\) is multiplied by itself once, resulting in \( a \times a \).What makes working with whole number exponents intuitive is that most of us naturally understand processes involving repetition. They relate to counting and basic whole number arithmetic. Remember:

• Whole numbers are numbers without fractions or decimals.
• They are always positive or zero in this context.
• Exponents apply to the base as many times as the exponent indicates.

Understanding how whole numbers interact in exponentiation lays the foundation needed for grasping more complex mathematical operations. This makes them essential for building basic mathematical literacy and competence.