Problem 81 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The single radical expression is \( \sqrt[6]{432} \).

Step 1: Express Each Radical with Rational Exponents

First, we express each radical using rational exponents. The square root \( \sqrt{3} \) can be written as \( 3^{1/2} \), and the cube root \( \sqrt[3]{4} \) can be written as \( 4^{1/3} \).

Step 2: Multiply the Expressions

Now multiply the expressions \( 3^{1/2} \) and \( 4^{1/3} \). This step involves applying the property of exponents that states \( a^{m} \cdot b^{n} = (a \cdot b)^{1} \). However, since they have different bases, they remain separate in multiplication: \( 3^{1/2} \cdot 4^{1/3} = (3^{1/2} \cdot 4^{1/3}) \).

Step 3: Write as a Single Radical

Combine the terms under a single radical by finding a common exponent. The least common multiple of 2 and 3 is 6. Rewrite the exponents with a common denominator of 6: \( 3^{1/2} = 3^{3/6} \) and \( 4^{1/3} = 4^{2/6} \). Now, \( 3^{3/6} \cdot 4^{2/6} = (3^3 \cdot 4^2)^{1/6} \).

Step 4: Simplify

Calculate \( 3^3 \) and \( 4^2 \). \( 3^3 = 27 \) and \( 4^2 = 16 \). Multiply those results: \( 27 \cdot 16 = 432 \). Thus, the expression becomes \( 432^{1/6} \).

Step 5: Conclude the Expression

The single radical expression is \( \sqrt[6]{432} \). This is the expression of the given radicals as a single radical expression.

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots and cube roots. These expressions are an essential part of algebra and come into play when dealing with polynomial equations. Understanding radical expressions is about interpreting and manipulating the values under the root symbol. For example, the square root of 3, written as \( \sqrt{3} \), and the cube root of 4, written as \( \sqrt[3]{4} \), are both radical expressions.
To combine or simplify these expressions, it's important to recognize the type of root being used—square root, cube root, and so on. Each type of root corresponds to a different radical expression.

• Square roots (like \( \sqrt{3} \)) reduce an expression by halves.
• Cube roots (like \( \sqrt[3]{4} \)) reduce an expression by thirds.

Understanding their formats lays the groundwork for more complex algebraic manipulations.

Rational Exponents

Rational exponents provide a convenient alternative to writing radical expressions. They are fractions where the numerator denotes the power, and the denominator indicates the root. For example, \( 3^{1/2} \) is a rational exponent form of \( \sqrt{3} \). Similarly, \( 4^{1/3} \) represents \( \sqrt[3]{4} \).
Recognizing these representations is crucial when simplifying or combining radical expressions. With rational exponents, algebraic manipulation becomes straightforward as they follow the usual arithmetic rules for exponents. Some useful aspects include:

• Converting radicals to rational expressions simplifies calculations.
• Understanding rational exponents helps in combining different radicals.

Using rational exponents is especially beneficial when attempting to unify multiple radicals into a single expression.

Properties of Exponents

The properties of exponents are foundational tools used to simplify expressions with powers. These properties help in converting and combining radical expressions by translating them into rational exponent form. For instance, multiplying two expressions with exponents, like \( 3^{1/2} \cdot 4^{1/3} \), follows specific rules. The properties of exponents that apply include:

• \( a^m \cdot a^n = a^{m+n} \), which applies if the bases are the same.
• The expressions remain separate when the bases differ, such as in \( 3^{1/2} \cdot 4^{1/3} \).

By understanding these properties, you can handle these expressions more effectively, simplifying them into more manageable forms.

Combining Radicals

Combining radicals involves unifying multiple radical expressions into a single radical. This requires understanding both radical expressions and rational exponents skills. To combine radicals, it often helps to find a common denominator for their rational exponents. For example, \( 3^{1/2} \) and \( 4^{1/3} \) were rewritten with a common denominator of 6 as \( 3^{3/6} \) and \( 4^{2/6} \).
This step allows you to express the combined radical as a single radical expression, such as \( (3^3 \cdot 4^2)^{1/6} \), simplifying further to \( \sqrt[6]{432} \). Here’s how you can approach combining radicals:

• Rewrite each radical using rational exponents with a common denominator.
• Multiply the expressions to combine the radicals.
• Simplify the resulting expression into a single radical form.

Properly combining radicals leads to more elegant and simplified expressions ready for further use in solving algebraic problems.