Problem 77 - Algebra A Combined Function

Question

Use rational expressions to write as a single radical expression. $$ \sqrt[3]{x} \cdot \sqrt[4]{x} \cdot \sqrt[8]{x^{3}} $$

Step-by-Step Solution

Answer

\( \sqrt[24]{x^{23}} \)

Step 1: Understanding the Problem

We need to combine the three radical expressions \( \sqrt[3]{x} \), \( \sqrt[4]{x} \), and \( \sqrt[8]{x^3} \) into a single radical expression. This requires expressing each radical using rational exponents.

Step 2: Converting Radicals to Rational Exponents

The expression \( \sqrt[n]{x^m} \) can be rewritten as \( x^{m/n} \). Let's apply this to each term:\[ \sqrt[3]{x} = x^{1/3} \]\[ \sqrt[4]{x} = x^{1/4} \]\[ \sqrt[8]{x^3} = x^{3/8} \].

Step 3: Multiplying the Rational Exponents

Combine the expressions by multiplying the exponents of \( x \):\[ x^{1/3} \cdot x^{1/4} \cdot x^{3/8} = x^{1/3 + 1/4 + 3/8} \].

Step 4: Finding a Common Denominator

To add the fractions \( \frac{1}{3} \), \( \frac{1}{4} \), and \( \frac{3}{8} \), we need a common denominator. The least common multiple of 3, 4, and 8 is 24. Rewrite each fraction:\[ \frac{1}{3} = \frac{8}{24}, \quad \frac{1}{4} = \frac{6}{24}, \quad \frac{3}{8} = \frac{9}{24} \].

Step 5: Adding the Rational Exponents

Add the rewritten fractions:\[ \frac{8}{24} + \frac{6}{24} + \frac{9}{24} = \frac{23}{24} \].

Step 6: Writing as a Single Radical Expression

Convert the expression back to a radical form. Since \( x^{23/24} \) is the result, rewrite it as a radical:\[ \sqrt[24]{x^{23}} \].

Key Concepts

Rational Exponents

Rational exponents are another way of expressing radicals. They allow for a different notation, which can be very useful in algebraic manipulations. When you see a radical such as \( \sqrt[n]{x^m} \), this can be expressed using a rational exponent as \( x^{m/n} \). This transformation is essential when working with radicals, especially when you need to perform operations like multiplication or division.
By converting radicals into rational exponents, calculations become similar to those involving ordinary exponents.

\item The exponent's numerator indicates the power to which the base is raised.
\item The denominator shows the root being taken of the base.

This conversion helps in simplifying expressions when dealing with multiple radicals.

Radicals

Radicals often appear daunting at first. However, they are simply ways of representing roots. The most common radical sign we encounter is the square root \( \sqrt{} \). But there are various types, including cube roots, fourth roots, and higher. If you see \( \sqrt[n]{x} \), it means you are looking for a number which, when raised to the power of \( n \), gives you \( x \).
In the exercise, we dealt with radicals like \( \sqrt[3]{x} \), \( \sqrt[4]{x} \), and \( \sqrt[8]{x^{3}} \). Understanding how to manipulate these radicals by converting them into rational exponents can make operations much simpler. Remember, the root gives a fraction of an exponent, aiding in ease when combining multiple expressions.

Common Denominator

Finding a common denominator is crucial when adding or comparing fractions. In the exercise, this was a necessary step to combine the rational exponents. To add fractions with different denominators, you need a common multiple. This is termed the "least common multiple" (LCM) and helps unify denominators.
For the fractions \( \frac{1}{3} \), \( \frac{1}{4} \), and \( \frac{3}{8} \), the LCM was 24. By rewriting each fraction with this common denominator, we make them compatible for addition:

\item \( \frac{1}{3} = \frac{8}{24} \) \item \( \frac{1}{4} = \frac{6}{24} \) \item \( \frac{3}{8} = \frac{9}{24} \)

With a unified base, you can easily combine the values to simplify the expression.

Multiplying Exponents

Multiplying expressions with the same base (in this case, \( x \)) involves adding their exponents. This is a vital property of exponents: \( x^a \times x^b = x^{a+b} \). In the exercise, we translated radicals into rational exponents and then combined them by adding the exponents. Through this property, even complex radical expressions can be reduced efficiently.
Consider \( x^{1/3} \), \( x^{1/4} \), and \( x^{3/8} \). These expressions, with the same base, allow for addition of exponents after finding a common denominator:

\item First, we transform them into fractions with a common denominator
\item Then, proceed with the addition: \( \frac{8}{24} + \frac{6}{24} + \frac{9}{24} = \frac{23}{24} \)

This results in \( x^{23/24} \), which further simplifies the process of expressing as a single radical.