Problem 66 - Algebra for College Students

Question

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(3 x^{\frac{1}{4}} y^{\frac{1}{5}}\right)^{3}\)

Step-by-Step Solution

Answer

The simplified expression is \(27x^{\frac{3}{4}}y^{\frac{3}{5}}\).

Step 1: Apply the Power of a Power Rule

The expression \( (3x^{\frac{1}{4}}y^{\frac{1}{5}})^3 \) indicates that each factor inside the parentheses is raised to the power of 3. According to the power of a power rule, \( (a^m)^n = a^{m \cdot n}\). Apply this rule to each term: \[3^3,\quad (x^{\frac{1}{4}})^3,\quad (y^{\frac{1}{5}})^3\].

Step 2: Simplify the Coefficient

Raise the coefficient to the power of 3: \(3^3 = 27\). This is straightforward since \(3 \times 3 \times 3 = 27\).

Step 3: Simplify the x-term

Apply the power rule \((x^{\frac{1}{4}})^3\): \(x^{\frac{1}{4} \cdot 3} = x^{\frac{3}{4}}\). This involves multiplying the exponents, \(\frac{1}{4} \times 3 = \frac{3}{4}\).

Step 4: Simplify the y-term

Apply the power rule \((y^{\frac{1}{5}})^3\): \(y^{\frac{1}{5} \cdot 3} = y^{\frac{3}{5}}\). Multiply the exponents, \(\frac{1}{5} \times 3 = \frac{3}{5}\).

Step 5: Combine Simplified Terms

Combine the simplified terms to get the final result: \(27x^{\frac{3}{4}}y^{\frac{3}{5}}\). Ensure all exponents are positive, which they already are in this expression.

Key Concepts

Power of a Power Rule

Let's explore the power of a power rule, an invaluable tool in simplifying expressions with exponents. When you encounter an expression like \((a^m)^n\), this rule helps by letting us write it as \(a^{m \cdot n}\). This means you multiply the two exponents together.

For example, if you have \((x^{\frac{1}{4}})^3\), apply the rule by multiplying the exponents: \(\frac{1}{4} \times 3\). This results in \(x^{\frac{3}{4}}\). It's like combining the power applied to a base multiple times into a single, unified power.

This technique helps in efficiently handling more complex expressions, especially when dealing with variables raised to a fraction or integers. Remember, each piece inside the parentheses is individually subjected to this rule, hence ensuring a simplified and more manageable form. This is exactly what we did in the original problem, applying it to both \(x\) and \(y\) terms separately.

Positive Exponents

In mathematics, keeping exponents positive isn't just a stylistic choice; it's often required for clarity and convention. Positive exponents are simpler to interpret and prevent ambiguity when working with expressions.

Whenever you need to simplify an expression, one key step is to ensure all exponents are positive. For instance, if during your work you end up with something like \(x^{-2}\), rewrite it as \(\frac{1}{x^2}\) so the exponent turns positive. In our solution, luckily, all exponents naturally turned out to be positive after applying the power of a power rule, which aligned perfectly with the requirements of most standard mathematical practices.

Keeping exponents positive is especially useful when expressions get shared or used elsewhere, ensuring seamless reading and further simplification if needed.

Fractional Exponents

Fractional exponents might look intimidating at first, but they unlock a unique flexibility in algebraic expressions. When you see something like \(a^{\frac{m}{n}}\), it essentially means the \(n\)th root of \(a\) raised to the \(m\)th power.

To demystify this with an example, observe \(x^{\frac{1}{4}}\). This is the fourth root of \(x\), which can also be expressed as \(\sqrt[4]{x}\). When you apply it in expressions, such exponents provide precision and succinctness, particularly useful in complex equations or calculus.

In the simplified problem, fractional exponents like \(x^{\frac{3}{4}}\) and \(y^{\frac{3}{5}}\) arose naturally, as a product of applying the power rules. They elegantly capture complex root and power relationships in compact formats, making calculations orderly and interpretations more straightforward.