Problem 52 - Intermediate Algebra

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{a^{1 / 4} a^{-1 / 2}}{a^{2 / 3}} $$

Step-by-Step Solution

Answer

\(\frac{1}{a^{11/12}}\)

Step 1: Combine the Exponents in the Numerator

First, combine the exponents of the variable \(a\) in the numerator. The expression in the numerator is \(a^{1/4} \cdot a^{-1/2}\). We add the exponents because the bases are the same:\[1/4 + (-1/2) = 1/4 - 1/2 = -1/4.\]So, the expression becomes \(a^{-1/4}\).

Step 2: Apply Division Property of Exponents

Next, apply the division property of exponents. The expression becomes \(\frac{a^{-1/4}}{a^{2/3}}\). When dividing like bases, subtract the exponent in the denominator from the exponent in the numerator:\[-1/4 - 2/3.\]To subtract these, find a common denominator. The common denominator of 4 and 3 is 12. So, convert the fractions: \(-1/4 = -3/12\) and \(2/3 = 8/12\).

Step 3: Perform the Subtraction of Exponents

Subtract the exponents found in Step 2:\[-3/12 - 8/12 = -11/12.\]Thus, the expression simplifies to \(a^{-11/12}\).

Step 4: Express with Positive Exponent

Since the exponent is negative, express it with a positive exponent by writing the base in the denominator:\[a^{-11/12} = \frac{1}{a^{11/12}}.\]This ensures the expression only has positive exponents.

Key Concepts

Simplifying Expressions

To simplify expressions, especially those involving exponents, it's crucial to understand how to handle these nifty mathematical symbols. Exponents tell you how many times you multiply a number by itself. The challenge often is to simplify these expressions to their most concise form.

Here's a breakdown of some key techniques:

• **Combining Like Terms:** To simplify, **combine like terms**. These are terms with the same base and exponent. If they have the same base and different exponents, you can usually add or subtract the exponents.
• **Following Order of Operations:** Always remember to follow the mathematical order of operations (PEMDAS/BODMAS) - parentheses, exponents, multiplication and division (from left to right), addition, and subtraction.
• **Handling Fractional Exponents:** With fractional exponents, the numerator indicates the power, and the denominator indicates the root. Simplifying involves finding common bases and handling the entire index efficiently.

Start by dealing with exponents inside any **parentheses** first and work outward. This top-down approach will help keep things organized and manageable.

Negative Exponents

Negative exponents can seem tricky, but they have a simple rule: they involve the reciprocal of the base. By converting negative exponents to positive exponents, expressions become easier to interpret. Let’s see how we can manage them:

• **Understanding Negative Exponents:** A negative exponent indicates an **inverse operation**. For example, for a base \(a\), \(a^{-n} = \frac{1}{a^n}\). This flips the base to the denominator if it's positive and cancels it if the base is negative.
• **Converting to Positive Exponents:** You need to rewrite expressions with only positive exponents, as this simplifies calculations and comparisons. Our goal is to express the entire base with positive exponents.

In our solution, once we simplified to \(a^{-11/12}\), it was essential to **flip** it, resulting in \(\frac{1}{a^{11/12}}\). This satisfies the rule of expressing the formula succinctly with positive exponents.

Division Property of Exponents

Exponents have a helpful property that makes division straightforward: the division property of exponents. This property allows you to simplify expressions by directly handling the exponents.

• **Division Property:** When you divide like bases, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\). This makes managing division involving exponents clear-cut and manageable.
• **Using a Common Denominator:** When working with fractional exponents, always find a common denominator first. It streamlines the subtraction process, making calculations efficient.
• **Simplification Process:** Start by applying the division property, then reduce the expression to have positive exponents only if needed. This tidies up the formula for clearer results.

In the exercise, subtracting the exponents \(-1/4 - 2/3\) involved converting to common denominators, simplifying to \(-11/12\), and finally adjusting to positive exponents. Mastery of the division property offers a solid foundation for simplifying these expressions.