Problem 42 - 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. $$\frac{7^{-1 / 3} 7 r^{-3}}{7^{2 / 3} r^{-2}}$$

Step-by-Step Solution

Answer

\(\frac{1}{r}\)

Step 1: Simplify the Exponents in the Numerator

The expression in the numerator is \(7^{-1/3} \times 7 \times r^{-3}\). Using the property of exponents \(a^m \times a^n = a^{m+n}\), simplify the expression involving \(7\):\[7^{-1/3} \times 7 = 7^{-1/3 + 1} = 7^{2/3}\]Now our numerator becomes \(7^{2/3} \times r^{-3}\).

Step 2: Simplify the Exponents in the Denominator

The denominator of the original expression is \(7^{2/3} \times r^{-2}\).

Step 3: Simplify the Whole Fraction

Now we simplify the fraction \(\frac{7^{2/3} \times r^{-3}}{7^{2/3} \times r^{-2}}\). By canceling out \(7^{2/3}\) in both the numerator and the denominator, we get:\[\frac{r^{-3}}{r^{-2}}\]

Step 4: Apply the Quotient Rule for Exponents

Using the property \(a^m / a^n = a^{m-n}\), simplify \(\frac{r^{-3}}{r^{-2}}\):\[r^{-3 - (-2)} = r^{-3 + 2} = r^{-1}\]

Step 5: Express with Positive Exponents

Finally, express \(r^{-1}\) with a positive exponent by writing it as the reciprocal:\[\frac{1}{r}\]

Key Concepts

Exponents

Exponents represent numbers that denote how many times a base is multiplied by itself. They are used in mathematics to simplify expressions, especially when dealing with large numbers. For example, instead of writing \(7 \times 7 \times 7\), we can simplify it using exponents as \(7^3\). This notation provides a concise way of expressing repeated multiplication. When working with exponents that have a fractional part, such as \(7^{1/2}\), it signifies taking a root of the base. Here, it indicates the square root of 7. The knowledge of exponents is fundamental when simplifying algebraic expressions.

Properties of Exponents

Understanding the properties of exponents is crucial for simplifying expressions efficiently. Here are a few key properties that are commonly used:

• Product of Powers: \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers: \(a^m / a^n = a^{m-n}\).
• Power of a Power: \((a^m)^n = a^{m \times n}\).
• Negative Exponent: \(a^{-n} = \frac{1}{a^n}\).
• Zero Exponent: \(a^0 = 1\), provided \(a eq 0\).

When you apply these rules, it becomes easier to manipulate and simplify expressions involving exponents. In the given exercise, the product and quotient rules are used systematically to simplify the expression from a complex form to an easily interpretable one with positive exponents.

Simplification of Expressions

Simplification of algebraic expressions involves reducing them to their simplest form. This process often requires combining like terms and applying various algebraic principles and properties. In expressions with exponents, such as those in the original exercise, simplification includes the reduction of powers using properties of exponents.
The exercise involves fraction operations, where terms in the numerator and denominator can be canceled out if they are similar. For instance, the base 7 terms in both parts got canceled leading to:\[ \frac{r^{-3}}{r^{-2}} \implies r^{-3 + 2} = r^{-1} \]The goal is to express the final result with positive exponents to align with standard algebraic form. So, \(r^{-1}\) is rewritten as \(\frac{1}{r}\), completing the simplification process. The ability to express results in their simplest form with positive exponents makes them more standard and easier to interpret.