Problem 45 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. Write your answers with only positive exponents. Assume that all variables represent positive real numbers. $$\frac{8 p^{-3}\left(4 p^{2}\right)^{-2}}{p^{-5}}$$

Step-by-Step Solution

Answer

\(\frac{1}{2p^2}\)

Step 1: Distribute the Negative Exponent

The expression given is \( 8p^{-3}(4p^2)^{-2} \). The negative exponent can be distributed into both terms inside the parenthesis: \((4p^2)^{-2} = 4^{-2} imes (p^2)^{-2}\). This simplifies to \(\frac{1}{4^2} \times \frac{1}{p^4} = \frac{1}{16p^4}\).

Step 2: Simplify the Numerator

Substitute the simplified expression from Step 1 back into the numerator: \(8p^{-3} \times \frac{1}{16p^4}\). This expression becomes \(\frac{8}{16} \times p^{-3} \times p^{-4}\) which simplifies to \(\frac{1}{2}p^{-7}\).

Step 3: Simplify the Denominator

The denominator of the expression is \(p^{-5}\). The expression then becomes \(\frac{\frac{1}{2}p^{-7}}{p^{-5}}\).

Step 4: Combine Exponents and Simplify

Apply the properties of exponents to combine \(\frac{1}{2}p^{-7}\) and \(p^{-5}\). This simplifies to \(\frac{1}{2}p^{-7+5}\), which results in \(\frac{1}{2}p^{-2}\).

Step 5: Write with Positive Exponents

Rewrite the expression \(\frac{1}{2}p^{-2}\) with positive exponents. This becomes \(\frac{1}{2} \times \frac{1}{p^2} = \frac{1}{2p^2}\).

Key Concepts

Negative Exponents

Negative exponents can sometimes be confusing, but they follow a simple rule. When you see a negative exponent, it means that the base should be taken as a reciprocal. In simpler terms, instead of multiplying the number, you divide. For example, if you have a negative exponent like \( p^{-3} \), it means \( \frac{1}{p^3} \). The rule for negative exponents can be summarized as \( a^{-n} = \frac{1}{a^n} \). This rule helps convert negative exponents into positive ones, making the expression easier to manage.
In the problem above, we encounter several negative exponents: \( p^{-3} \), \( (4p^2)^{-2} \), and \( p^{-5} \). By applying the negative exponent rule, we can rewrite each term with positive exponents. This step is crucial because it simplifies the expressions and makes further calculations more straightforward.

Properties of Exponents

Dealing with exponents involves certain mathematical properties that allow us to simplify expressions. These properties apply not only to positive exponents but also hold true for negative ones. Here are the main properties of exponents:

• Product of Powers: \( a^m \times a^n = a^{m+n} \). This means when you multiply like bases, you add their exponents.
• Power of a Power: \( (a^m)^n = a^{m \times n} \). This means when raising a power to another power, you multiply the exponents.
• Power of a Product: \( (ab)^n = a^n \times b^n \). This property applies the exponent to both parts of the product.

In the example provided, these properties are used to simplify terms. For instance, \( (4p^2)^{-2} \) was split into \( 4^{-2} \) and \( (p^2)^{-2} \), following the power of a product property. Later, using the product of powers property, we consolidated the exponents of \( p \) to simplify further.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra that makes working with complex expressions much more manageable. It involves using exponent rules to rewrite expressions in simpler forms without changing their values. You begin by dealing with exponent rules first, like solving negative exponents to make all exponents in the problem positive.
Let's look back at the example: you start with \( 8p^{-3}(4p^2)^{-2} \). By applying properties of exponents and rewriting negative exponents, the expression becomes easier to handle.

• Calculate individual terms: Break down the expression into manageable parts, applying exponent rules.
• Combine terms: Use arithmetic operations to consolidate like terms, ensuring all exponents are positive as per the original problem's requirement.
• Simplify fully: Make sure the final expression is in its simplest form. Here we arrived at \( \frac{1}{2p^2} \), with all exponents positive.

Simplifying is like clearing up a puzzle, ensuring that all pieces fit neatly, resulting in a clearer and simpler expression without changing the mathematical meaning.