Problem 48 - Elementary and Intermediate Algebra

Question

Use the product and quotient rules for exponents to simplify each expression. $$ \frac{b^{4} b^{5}}{b^{2} b^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \(b^4\).

Step 1: Apply Product Rule to the Numerator

The numerator of the expression is given by \(b^4 b^5\). According to the product rule of exponents, when multiplying two powers that have the same base, you keep the base and add the exponents. Therefore, \(b^4 b^5 = b^{4+5} = b^9\).

Step 2: Apply Product Rule to the Denominator

The denominator of the expression is given by \(b^2 b^3\). Using the product rule again, you add the exponents: \(b^2 b^3 = b^{2+3} = b^5\).

Step 3: Simplify Using Quotient Rule

The simplified expression from Steps 1 and 2 is \(\frac{b^9}{b^5}\). According to the quotient rule of exponents, when dividing two powers with the same base, you keep the base and subtract the exponents: \(\frac{b^9}{b^5} = b^{9-5} = b^4\).

Step 4: Verify the Simplification

To ensure that the final result \(b^4\) is correct, check each step for arithmetic and application of the exponent rules. Everything adds up correctly, so the simplification is verified.

Key Concepts

Understanding the Product Rule of Exponents

The product rule of exponents is a handy rule that simplifies multiplying numbers or variables with exponents, as long as the bases are the same. It's as simple as keeping the base the same and adding up the exponents. For instance, if you have two terms like \( b^m \) and \( b^n \), the result of multiplying them would be \( b^{m+n} \).

This rule helps to make computations easier, especially when dealing with large numbers or multiple variables. Suppose you encounter \( b^4 \times b^5 \), just follow the product rule: \( b^{4+5} = b^9 \).

You can think of it this way: If you have a base that's repeated four times and another that's repeated five times, altogether it repeats nine times. This illustrates the power of adding exponents when the base is constant.

Mastering the Quotient Rule of Exponents

The quotient rule is another important tool when managing expressions with exponents. This rule is used when dividing terms with exponents that share the same base. According to this rule, you keep the base the same and subtract the exponent in the denominator from the one in the numerator.

For example, when you have an expression like \( \frac{b^{m}}{b^{n}} \), you simply need to calculate \( b^{m-n} \).

Using the provided example, after applying the product rule, you might end up with \( \frac{b^9}{b^5} \). Using the quotient rule means subtracting 5 from 9, which results in \( b^{4} \). This simplifies complex expressions and is based on the concept that dividing powers is akin to removing them in the count.

Techniques for Simplifying Exponential Expressions

Simplifying expressions with exponents can initially seem daunting, but by using rules like product and quotient, it becomes manageable.

Here’s a quick guide to avert confusion when simplifying these expressions:

• Identify expressions with the same base and use the product rule for multiplication.
• For division, apply the quotient rule to subtract exponents with the same base.
• Ensure to process each step distinctly for clarity, so mistakes are avoided.

Given the expression \( \frac{b^4 b^5}{b^2 b^3} \), first, simplify the numerator: \( b^4 \times b^5 = b^9 \). Then handle the denominator: \( b^2 \times b^3 = b^5 \).

Finally, apply the quotient rule: \( \frac{b^9}{b^5} \) to get \( b^{4} \). By methodically applying these rules, you keep the expression clear and the math precise.