Problem 59 - A Graphical Approach to College Algebra

Question

Find each product. Assume that all variables represent positive real numbers. $$\left(r^{1 / 2}-r^{-1 / 2}\right)^{2}$$

Step-by-Step Solution

Answer

The expanded expression is \( r - 2 + r^{-1} \).

Step 1: Identify the Binomial

Recognize \( (r^{1/2} - r^{-1/2})^2 \)as a binomial square, which can be expanded using the formula \((a-b)^2 = a^2 - 2ab + b^2\).

Step 2: Square the First Term

The first term is \( r^{1/2} \). To find its square, calculate:\[ (r^{1/2})^2 = r^{1/2 imes 2} = r^1 = r \]

Step 3: Square the Second Term

The second term is \( r^{-1/2} \). To find its square, calculate:\[ (r^{-1/2})^2 = r^{-1/2 imes 2} = r^{-1} \]

Step 4: Calculate the Product of the Terms and Double It

The terms are \( r^{1/2} \) and \( r^{-1/2} \). Find their product and multiply by 2:\[ 2 \cdot (r^{1/2} \cdot r^{-1/2}) = 2 \cdot r^{1/2-1/2} = 2 \cdot r^{0} = 2 \cdot 1 = 2 \]

Step 5: Combine All Parts Together

Combine all terms from the previous steps:\[ r - 2 + r^{-1} \]This is the expanded form of the original expression.

Key Concepts

Rational Exponents

In mathematics, rational exponents are a way to express roots and powers using fractions. Understanding rational exponents helps simplify expressions and solve equations where variables are under a root or power. Consider an expression like \(r^{1/2}\), which represents \(\sqrt{r}\) or the square root of \(r\). The numerator signifies a root, while the denominator determines what power the base is raised to. For instance:

• \(r^{1/2} = \sqrt{r}\)
• \(r^{3/2} = (\sqrt{r})^3\)
• \(r^{-1/2} = \frac{1}{\sqrt{r}}\)

Rational exponents offer a compact way to handle complex roots and powers within algebraic expressions. They are essential for simplifying products, quotient, and powers and are especially helpful in exponential equations where multiple roots are involved. Understanding them lays the groundwork for later mathematical concepts.

Exponent Rules

Exponent rules, also known as the laws of exponents, provide the guidelines for simplifying expressions with exponents. When dealing with powers and roots, these rules make calculations straightforward. Here are some fundamental exponent rules:

• Power of a Power Rule: \((a^m)^n = a^{m \times n}\)
• Product of Powers Rule: \(a^m \times a^n = a^{m+n}\)
• Quotient of Powers Rule: \(\frac{a^m}{a^n} = a^{m-n}\)
• Zero Exponent Rule: For any non-zero number \(a\), \(a^0 = 1\)
• Negative Exponent Rule: \(a^{-m} = \frac{1}{a^m}\)

In the context of the exercise, we apply these rules while expanding the binomial expression \((r^{1/2} - r^{-1/2})^2\). Understanding these rules makes the expansion process simpler and aids in validating each step, ensuring accuracy.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (like addition and multiplication). Expressions such as \(r - 2 + r^{-1}\) are made up of terms that are often simplified or expanded using algebraic techniques. It's essential to be well-versed in manipulating these expressions to solve equations and simplify complex terms.An algebraic expression can be transformed using various methods, including:

• Using the distributive property to expand or factor
• Simplifying by combining like terms
• Applying exponent rules to reorganize terms

In the original exercise, we are expanding and simplifying \((r^{1/2} - r^{-1/2})^2\). Each term is carefully calculated using exponent rules, before being combined into a simplified expression, \(r - 2 + r^{-1}\). These techniques are vital for moving smoothly between different forms of an expression, leading to simplified solutions in algebra.