Problem 60 - Elementary and Intermediate Algebra

Question

Use the product and power rules for exponents to simplify each expression. $$ \left(y^{3} y^{4}\right)^{4} $$

Step-by-Step Solution

Answer

The simplified expression is \(y^{28}\).

Step 1: Apply the Power Rule

The expression is \( \left(y^{3} y^{4}\right)^{4} \). First, apply the power rule \((a^m)^n = a^{m \cdot n}\) to simplify the expression inside the parentheses. Combine the exponents for \(y^{3} y^{4}\) by adding them: \(y^{3+4} = y^{7}\). Now the expression is simplified to \( \left(y^{7}\right)^{4} \).

Step 2: Apply the Power Rule Again

Now apply the power rule \((y^m)^n = y^{m \cdot n}\) to the expression \(\left(y^{7}\right)^{4}\). Multiply the exponents: \(7 \cdot 4 = 28\). Thus, the expression becomes \(y^{28}\).

Key Concepts

Product Rule

The product rule for exponents makes simplifying expressions with the same base much easier. When you multiply two exponents with the same base, you simply add the exponents. In the expression \( y^3 \times y^4 \), both terms have the base \( y \). This tells us it's time to use the product rule. So, you add the exponents 3 and 4 together. The calculation looks like this: \( y^{3+4} = y^7 \). Breaking it down, the product rule operates under this principle:

• \( a^m \times a^n = a^{m+n} \)

Knowing how to apply this rule will save you time and make working with expressions with the same base straightforward.

Power Rule

The power rule is another handy tool in simplifying expressions with exponents. If you have an exponent raised to another exponent, you multiply the exponents together. This is often seen in expressions like \((a^m)^n\), and here's how it works:

• The formula is: \((a^m)^n = a^{m \cdot n}\)

Let's see it in action with our expression. Initially, we had \( \left(y^3 y^4\right)^4 \). After using the product rule, it became \( \left(y^7\right)^4 \). Using the power rule here means multiplying 7 and 4. The calculation is \( y^{7 \cdot 4} = y^{28} \).
The power rule allows you to transform otherwise complex expressions into simpler, single terms.

Simplifying Expressions

Simplifying expressions using exponents involves using rules like the product and power rules. These rules help to consolidate complex expression into simpler forms by reducing the number of terms and making calculations easier. When faced with an expression such as \( \left(y^3 y^4\right)^4 \), it's all about applying these rules systematically.
Start by identifying terms with the same base and use the product rule first. Here it's \( y^3 \) and \( y^4 \), which simplify to \( y^7 \). Next, use the power rule. As we saw:

• \( \left(y^7\right)^4 \) becomes \( y^{28} \)

By knowing the steps and applying the right rules, you can handle any expression with ease. Simplifying expressions is not just about memorizing rules—it's also understanding how and when to apply them, which clears the path to solving more complex equations later on.