Problem 84 - Algebra A Combined Function

Question

Use rational expressions to write as a single radical expression. $$ \sqrt[4]{5} \cdot \sqrt[3]{x} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \sqrt[12]{125x^4} \).

Step 1: Identify the Root Powers

The expression includes two different roots: the fourth root of 5 and the cube root of x. These can be rewritten using rational exponents. Specifically, \( \sqrt[4]{5} \) can be expressed as \( 5^{1/4} \), and \( \sqrt[3]{x} \) can be expressed as \( x^{1/3} \).

Step 2: Multiply the Expressions

Use the property of multiplying powers with the same base, which states that when you multiply expressions with the same base, you add the exponents. However, here we have different bases. The expression becomes \( 5^{1/4} \cdot x^{1/3} \).

Step 3: Express as a Single Radical

The goal is to combine into a single radical expression, meaning we need a common power for the radical. To do this, we find a common denominator for the exponents, which is 12. Thus, we rewrite \( 5^{1/4} \) as \( 5^{3/12} \) and \( x^{1/3} \) as \( x^{4/12} \). This gives us \( 5^{3/12} \cdot x^{4/12} = (5^3 \cdot x^4)^{1/12} \).

Step 4: Write as a Single Radical Expression

The expression \( (5^3 \cdot x^4)^{1/12} \) is in the form of a single radical expression. Simplifying \( 5^3 \) gives 125, leading to \( \sqrt[12]{125x^4} \). This is a single radical expression.

Key Concepts

Rational Exponents

Rational exponents are a useful way to express roots in the form of an exponent. This method allows us to apply the rules of exponents consistently and simplify complex expressions. When you see a radical such as \(\sqrt[n]{a}\), it can be rewritten as \(a^{1/n}\). The numerator indicates the power, while the denominator represents the root. For example, in the expression \(\sqrt[4]{5}\), the fourth root of 5 is expressed as \(5^{1/4}\). Similarly, \(\sqrt[3]{x}\) becomes \(x^{1/3}\). This transformation into rational exponents helps make calculations involving radicals much easier and more straightforward.

Multiplying Radicals

Multiplying radicals involves using specific rules to combine them effectively. When multiplying radicals with the same base, you can add the exponents as per exponent rules. However, if the radicals have different bases, each radical is treated separately. In our example, we have \(\sqrt[4]{5} \cdot \sqrt[3]{x}\), which translates to \(5^{1/4}\cdot x^{1/3}\) using rational exponents. Even though you can't combine the bases directly, this step is integral when eventually expressing in a single radical form. Remember, when multiplying different radical bases, you maintain each base but focus on aligning the powers.

Common Denominator in Exponents

Achieving a common denominator for exponents is essential to express products of different radical bases as a single radical. In our case, we have the exponents \(1/4\) and \(1/3\), representing different root powers. To combine them, we convert these fractions to have a common denominator. The least common denominator of 4 and 3 is 12. Therefore, \(5^{1/4}\) becomes \(5^{3/12}\), and \(x^{1/3}\) becomes \(x^{4/12}\). This rewriting allows us to express the original product in terms of a unified power of \(1/12\). This step is pivotal for seamlessly combining the radicals into a single expression.

Single Radical Expression

The ultimate goal in simplifying radical expressions is to consolidate them into a single radical form. A single radical form is often more intuitive and simpler to work with or evaluate. Once you have adjusted the exponents using a common denominator, you can translate them into a single radical. In our example, \(5^{3/12} \cdot x^{4/12}\) becomes \((5^3 \cdot x^4)^{1/12}\). With further simplification of \(5^3\) to 125, we get \(\sqrt[12]{125x^4}\). With this expression, you are now viewing the entire operation under one radical, simplifying calculations and understanding of the form.