Problem 65 - A Graphical Approach to College Algebra

Question

Factor, using the given common factor. Assume that all variables represent positive real numbers. $$p^{-3 / 4}-2 p^{-7 / 4} ; \quad p^{-7 / 4}$$

Step-by-Step Solution

Answer

The expression factorized is \( p^{-7/4} (p - 2) \).

Step 1: Identify the Common Factor

In the expression \( p^{-3/4} - 2p^{-7/4} \), it is given that \( p^{-7/4} \) is the common factor. To factor out \( p^{-7/4} \), each term in the expression must include \( p^{-7/4} \) as a factor.

Step 2: Divide Each Term by the Common Factor

To factor out \( p^{-7/4} \), divide each term of the expression by this common factor.- Divide the first term: \( p^{-3/4} \div p^{-7/4} = p^{(-3/4) - (-7/4)} = p^{4/4} = p^1 = p \).- Divide the second term: \(-2p^{-7/4} \div p^{-7/4} = -2\), because \( p^{-7/4} \div p^{-7/4} = 1 \).

Step 3: Write the Factored Expression

Now that each term has been divided by \( p^{-7/4} \), we can write the expression as: \[ p^{-3/4} - 2p^{-7/4} = p^{-7/4} (p - 2) \].

Key Concepts

Common Factor

When you look at expressions like \( p^{-3/4} - 2p^{-7/4} \), you might notice repeated elements. These repeated elements are often known as the common factor. In this example, the common factor is given as \( p^{-7/4} \). Remember, a common factor is any number or term that divides all the terms of the expression without leaving a remainder. This is very useful when simplifying expressions, just like pulling out a common ingredient from a recipe. Each term of the expression should share this factor, which feels a bit like finding what all the parts have in common.

• A common factor makes expressions easier to manage.
• It helps reduce complex expressions into simpler ones.
• Identifying the common factor is the first step in many algebraic processes.

In factoring, once you find the common factor, you can "pull it out" of the original expression, just as you would rearrange the furniture in a room to make it feel more spacious and organized.

Exponents

Exponents can initially seem tricky, but they are super handy once you understand how they work. In simple terms, an exponent tells you how many times to "multiply" a number by itself. For instance, an exponent of \( -7/4 \) in the expression \( p^{-7/4} \) indicates a fractional power and a negative exponent.

• Fractional exponents are another way of expressing roots. For example, \( p^{1/2} = \sqrt{p} \).
• Negative exponents don't mean to be scared, but rather they signify taking the reciprocal. So, \( p^{-n} = \frac{1}{p^n} \).
• When performing operations with exponents, the rules of addition and subtraction apply to the exponents themselves.

Using exponent rules, even complexities can be broken down, which can help in transforming and simplifying expressions efficiently. Let's remember it works almost like language in mathematics, providing concise instructions on how to handle numbers.

Positive Real Numbers

When working with expressions involving exponents and factors, it’s key to note that we're assuming all variables are positive real numbers. This is important because real numbers can be any number that exists on the number line, which includes fractions and decimals, not just whole numbers.

• Real numbers are used because they cover "real" quantities that appear in mathematics, like measures of length and volume.
• Positive denotes numbers greater than zero, ensuring clarity, especially when dealing with roots or fractional exponents.
• Incidents like roots of negative numbers become clearer because under these assumptions, they are simply not possible.

Dealing only with positive numbers here helps prevent scenarios that could complicate our mathematical work, like undefined operations or imaginary numbers. This makes all arithmetic operations smooth sailing, much like ensuring all your equations remain in the sunny realm of reality. Keeping this assumption simplifies not just your solution, but also the steps you take to get there.