Problem 47 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\(y^4\)

Step 1: Apply the Product Rule of Exponents in the Numerator

The product rule of exponents states that when you multiply two expressions with the same base, you add the exponents. In the numerator, apply the product rule to \(y^3\) and \(y^4\). This gives:\[y^{3 + 4} = y^7\]

Step 2: Simplify the Denominator

In the denominator, you have \(y \times y^2\). Apply the product rule of exponents to simplify this. This gives:\[y^{1 + 2} = y^3\]

Step 3: Apply the Quotient Rule of Exponents

Now, you have \(\frac{y^7}{y^3}\). The quotient rule of exponents states that when you divide two expressions with the same base, you subtract the exponents. Apply the quotient rule:\[y^{7 - 3} = y^4\]

Key Concepts

Product Rule for Exponents

The product rule for exponents is incredibly useful when dealing with expressions that have the same base. This rule states that when you multiply two exponential expressions with the same base, you simply add their exponents. Here's the formula:

\[a^m \times a^n = a^{m+n}\]This rule simplifies calculations and is a foundational concept in algebra.

Let's take an example to see the product rule in action. Consider the expression \(y^3 \times y^4\). Since the base \(y\) is the same in both terms, you add the exponents 3 and 4. Therefore, \(y^3 \times y^4\) simplifies to \(y^{3+4} = y^7\).

By using this straightforward rule, we make complex-looking expressions much easier to handle.

Quotient Rule for Exponents

The quotient rule for exponents helps when you're working with divisions of the same base. According to this rule, when you divide two expressions with the same base, you subtract the exponent of the denominator from the exponent of the numerator. The formula is straightforward:

\[a^m \div a^n = a^{m-n}\]This rule is very efficient in simplifying expressions that might otherwise look intimidating.

Consider the expression \(\frac{y^7}{y^3}\). Using the quotient rule, you subtract the exponent in the denominator (3) from the exponent in the numerator (7):

\[y^{7-3} = y^4\]Applying this rule cuts down the complexity by eliminating the denominator and leaving a single, simpler expression.

Simplifying Expressions with Exponents

Simplifying expressions with exponents involves using rules like the product and quotient rules we previously discussed. It helps in reducing the size of expressions, making them easier to understand and work with.

Start by looking for opportunities to apply the product rule. Combine like bases in the numerator first, as seen with \(y^3 \times y^4\) simplifying to \(y^7\). Similarly, simplify the denominator, \(y \times y^2\), to \(y^3\).

After addressing both the numerator and the denominator, use the quotient rule to simplify the overall fraction. Here, \(\frac{y^7}{y^3}\) becomes \(y^4\).

By layering these steps, you can efficiently transform a complex-looking expression into a much simpler form. Remember, these rules are key in making the work with exponents manageable and are valuable tools in your algebra arsenal.