Problem 35 - 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. $$\left(2 y^{3 / 4} z\right)\left(3 y^{-2} z^{-1 / 3}\right)$$

Step-by-Step Solution

Answer

6z^{2/3}/y^{5/4}

Step 1: Identify the Expression

Examine the expression to be simplified: \(\left(2 y^{3/4} z\right)\left(3 y^{-2} z^{-1/3}\right)\). Notice that it involves multiplication of terms with exponents.

Step 2: Distribute the Multiplication

Multiply the coefficients and like bases separately. First, multiply the coefficients: \(2 \times 3 = 6\).

Step 3: Apply Exponent Rules for 'y'

Combine the exponents of \(y\) by adding them: \(y^{3/4} \cdot y^{-2} = y^{3/4 + (-2)} = y^{3/4 - 8/4} = y^{-5/4}\).

Step 4: Apply Exponent Rules for 'z'

Similarly, add the exponents for \(z\): \(z \cdot z^{-1/3} = z^{1 + (-1/3)} = z^{1 - 1/3} = z^{2/3}\).

Step 5: Write the Expression with Positive Exponents

Since the exponent of \(y\) is negative, rewrite \(y^{-5/4}\) as \(\frac{1}{y^{5/4}}\). The final expression is: \[ 6 \cdot \frac{z^{2/3}}{y^{5/4}} \]

Key Concepts

Exponent Rules

Exponent rules are mathematical guidelines that help you work efficiently with powers. When you multiply expressions with the same base, you add their exponents. Similarly, when dividing, you subtract the exponents of like bases. These rules simplify complex algebraic expressions and make calculations much easier.

• Multiplication of Like Bases: For any base \(a\), the rule is \(a^m \cdot a^n = a^{m+n}\).
• Division of Like Bases: Similarly, \(\frac{a^m}{a^n} = a^{m-n}\).

When dealing with coefficients (the numbers in front of variables), you multiply them separately from the variables. In the given example, for instance, 2 and 3 were multiplied to give 6, while the exponents of like bases were operated on using the above rules. Understanding these rules allows for quick simplification of expressions.

Positive Exponents

Positive exponents indicate how many times a number is multiplied by itself. They can often make calculations simpler and more intuitive. In mathematics, it is conventional to express the final result of polynomial simplifications using positive exponents.
To convert a negative exponent into a positive one, you invert the base. For example, \(x^{-n} = \frac{1}{x^n}\). This concept was evident when the expression \(y^{-5/4}\) was transformed into a more understandable form \(\frac{1}{y^{5/4}}\). Keeping exponents positive not only simplifies calculations but also aligns with common mathematical practices.

Simplifying Expressions

The process of simplifying expressions involves reducing them to their simplest form. This generally means removing complex fractions or negative exponents, and combining like terms to make the expression more manageable.

• Combining Like Terms: Use exponent rules to combine terms that have the same base.
• Removing Negative Exponents: Convert any negative exponents into positive ones by rewriting them as fractions.

In the problem provided, after distributing and using exponent rules to simplify, the expression was written in a polished form with positive exponents: \(6 \cdot \frac{z^{2/3}}{y^{5/4}}\). Simplification is key for easier computation and interpretation of algebraic expressions.