Problem 16 - Algebra 2

Question

If \(a\) is positive, then \(\frac{a^{5} \cdot a^{\frac{2}{3}}}{a^{\frac{4}{3}}}=?\) $$ \begin{array}{llll}{\text { A } a} & {\text { B } a^{2}} & {\text { C } \sqrt[3]{a^{13}}} & {\text { D } \sqrt[3]{a^{3}}}\end{array} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\sqrt[3]{a^{13}}\), which corresponds to option C.

Step 1: Simplify the Numerator

The numerator of the expression \(a^{5} \cdot a^{\frac{2}{3}}\) can be simplified by using the property of exponents that states \(a^{m} \cdot a^{n} = a^{m+n}\). Therefore, we add the exponents: \(5 + \frac{2}{3}\). To add these, convert 5 into a fraction: \(5 = \frac{15}{3}\). Now add: \(\frac{15}{3} + \frac{2}{3} = \frac{17}{3}\). Thus, the numerator becomes \(a^{\frac{17}{3}}\).

Step 2: Write the Full Expression with Simplified Numerator

Now that the numerator is simplified to \(a^{\frac{17}{3}}\), rewrite the original expression as \(\frac{a^{\frac{17}{3}}}{a^{\frac{4}{3}}}\).

Step 3: Simplify the Equation Using Quotient Rule

Apply the quotient rule, \(\frac{a^{m}}{a^{n}} = a^{m-n}\), to the expression. Subtract the exponents \(\frac{17}{3} - \frac{4}{3}\). This gives us \(\frac{17}{3} - \frac{4}{3} = \frac{13}{3}\). Therefore, the expression simplifies to \(a^{\frac{13}{3}}\).

Step 4: Convert the Result to a Suitable Form

The problem asks for a multiple-choice answer format. Since \(a^{\frac{13}{3}} = \sqrt[3]{a^{13}}\), compare this with the choices given: A: \(a\), B: \(a^{2}\), C: \(\sqrt[3]{a^{13}}\), D: \(\sqrt[3]{a^{3}}\). Hence, the correct answer corresponding to our simplification is option C: \(\sqrt[3]{a^{13}}\).

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial for simplifying expressions involving powers. In this exercise, the rule that states \(a^m \cdot a^n = a^{m+n}\) is applied. This is known as the multiplication rule of exponents. When you multiply numbers with the same base, you add their exponents.
For instance, in the expression \(a^5 \cdot a^{\frac{2}{3}}\), you add 5 (or \(\frac{15}{3}\) when expressed as a fraction for consistency) to \(\frac{2}{3}\), resulting in \(a^{\frac{17}{3}}\).
Here are some key properties of exponents:

• Multiplication of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Division of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)
• Zero Exponent Rule: \(a^0 = 1\) for any non-zero \(a\)

Mastering these rules simplifies the process of working with expressions involving exponents.

Simplifying Expressions

Simplifying expressions often involves applying exponent rules like those discussed earlier. In the step-by-step solution of this problem, the expression \(a^5 \cdot a^{\frac{2}{3}}\) is the numerator. After applying the multiplication rule, we get a simplified numerator \(a^{\frac{17}{3}}\).
Next, we express the overall fraction as \(\frac{a^{\frac{17}{3}}}{a^{\frac{4}{3}}}\). By applying the quotient rule \(\frac{a^m}{a^n} = a^{m-n}\), we subtract the exponents.
This subtraction simplifies to \(\frac{17}{3} - \frac{4}{3} = \frac{13}{3}\). Thus, the original complex-looking expression is now much simpler: \(a^{\frac{13}{3}}\).
Simplification is essential for solving algebraic problems efficiently, which involves breaking down complex problems into manageable parts using mathematical rules.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both are algebraic expressions. Simplifying algebraic fractions often involves using the properties of exponents as shown earlier.
In our example, the fraction's numerator and denominator are expressions with the base \(a\). To simplify \(\frac{a^{\frac{17}{3}}}{a^{\frac{4}{3}}}\), understanding that each part must be dealt with using exponent rules is crucial.
By applying the quotient rule, apparent complexity is greatly reduced. The result \(a^{\frac{13}{3}}\) can be rewritten in different forms depending on the context or problem requirement, such as \(\sqrt[3]{a^{13}}\) for clarity, which was necessary in this exercise to match a multiple-choice question format.
Effective handling of algebraic fractions is necessary for simplifying expressions and for higher-level algebra tasks, ensuring that solutions meet given answer formats or forms.