Problem 49 - Elementary and Intermediate Algebra

Question

Use the product and quotient rules for exponents to simplify each expression. $$ \frac{a^{2} a^{3} a^{4}}{a^{8}} $$

Step-by-Step Solution

Answer

The simplified expression is \(a\).

Step 1: Apply the Product Rule

To simplify the expression \(a^2 a^3 a^4\), we use the product rule for exponents, which states that when multiplying like bases, we add their exponents. Thus, \(a^2 a^3 a^4 = a^{2+3+4} = a^9\).

Step 2: Apply the Quotient Rule

Now simplify \(\frac{a^9}{a^8}\) using the quotient rule, which states that when dividing like bases, we subtract their exponents. So, \(\frac{a^9}{a^8} = a^{9-8} = a^1\).

Step 3: Simplify the Expression

The simplified form of \(a^1\) is just \(a\), since any number (or variable) raised to the power of 1 is itself.

Key Concepts

Product Rule for Exponents

The product rule for exponents simplifies expressions with the same base. When you multiply terms with the same base, you can add the exponents. Think of it as a shortcut to make things easier. For instance, if you have \(a^2 \times a^3 \times a^4\), the product rule lets you add up the exponents: \(2 + 3 + 4 = 9\). So, \(a^2 \times a^3 \times a^4 = a^9\).
This rule helps you manage large numbers of multiplied terms without writing them out fully. By using it, you’re effectively turning a potentially cumbersome multiplication into a simple addition.

Quotient Rule for Exponents

After mastering the product rule, the quotient rule for exponents is your next best friend! The quotient rule works with division and simplifies expressions by subtracting exponents when the bases are identical. For instance, when you encounter the expression \(\frac{a^9}{a^8}\), you can subtract the exponents: \(9 - 8 = 1\). Therefore, \(\frac{a^9}{a^8} = a^1\).
This is quite handy when simplifying fractions with like bases and helps avoid confusion over more complex divisions. Always ensure that the bases are the same to apply this rule effectively.

Simplifying Expressions with Exponents

Simplifying expressions with exponents involves reducing them to their simplest form. The previously discussed rules are key tools in this process. Let’s say you've already applied the product and quotient rules. After using these rules in expressions such as \(\frac{a^9}{a^8}\), you arrive at \(a^1\).
The final step in simplifying is recognizing that any base to the power of one remains unchanged. So, \(a^1 = a\), and this is as simple as it gets!

• Always check your final expression to ensure no further simplification is possible.
• The less cluttered your expression, the easier it is to work with in further problems.

By applying these strategies, you'll be able to simplify complex exponential expressions with ease.