Problem 61 - Elementary and Intermediate Algebra

Question

Use the product and power rules for exponents to simplify each expression. $$ \left(p^{2} p^{3}\right)^{5} $$

Step-by-Step Solution

Answer

The expression simplifies to \(p^{25}\).

Step 1: Apply the Product Rule

According to the product rule, when multiplying like bases, you add the exponents. For the expression \(p^{2} p^{3}\), we add the exponents: \(2 + 3 = 5\). Hence, the expression simplifies to \(p^{5}\).

Step 2: Raise to the Power

Now, we need to raise \(p^5\) to the power of 5. By the power of a power rule, which states that \((a^m)^n = a^{m imes n}\), we calculate \((p^5)^5\). Multiply the exponents: \(5 \times 5 = 25\).

Step 3: Final Simplification

This means \((p^5)^5\) simplifies to \(p^{25}\). There are no further calculations necessary because we have simplified the expression as much as possible.

Key Concepts

Product Rule for Exponents

When you're working with exponents, you'll often encounter expressions where you need to multiply similar bases. This is where the product rule for exponents becomes very handy. The rule states that when you multiply like bases, you simply add the exponents.

For example, consider the expression \(p^2 \times p^3\). Here, "like bases" means both terms have the base \(p\). According to the product rule, we add the exponents: \(2 + 3 = 5\). So, \(p^2 \times p^3 = p^5\).

To apply this rule correctly:

• Ensure the bases are the same.
• Add the exponents together.
• This simplifies your expression significantly.

The beauty of this rule lies in its simplicity and its ability to turn long expressions into much neater forms.

Power Rule for Exponents

The power rule for exponents comes into play when you have a power raised to another power. This rule states that when you take an exponent to another exponent, you multiply the exponents.

Let’s take a deeper look using our problem. You have \((p^5)^5\). Here, you're applying the power rule. Multiply the exponents: \(5 \times 5 = 25\), giving you \(p^{25}\).

Follow these steps:

• Identify the base, which in our example is \(p\).
• Multiply the inner and outer exponents.
• Simplify by computing this product, which gives the final exponent.

Using the power rule can make complex exponent problems much easier by reducing expressions efficiently.

Simplifying Expressions

Simplifying expressions with exponents involves reducing the expression to its easiest form without changing its value. This is where both the product and power rules help to streamline calculations.

For our example, we started with \((p^2 p^3)^5\). First, apply the product rule: \(p^2 \times p^3\) simplifies to \(p^5\). Next, use the power rule: \((p^5)^5\) further simplifies to \(p^{25}\).

Here's a process to help simplify efficiently:

• Apply the product rule when multiplying like bases.
• Use the power rule for expressions raised to an additional power.
• Simplify step-by-step, ensuring clarity at each stage before moving to the next.

With practice, these rules and methods make simplifying much more manageable.