Problem 94 - Intermediate Algebra

Question

Use rational expressions to write as a single radical expression. $$ \sqrt[3]{5} \cdot \sqrt{2} $$

Step-by-Step Solution

Answer

\( \sqrt[6]{200} \)

Step 1: Understand the Problem

We need to combine two radical expressions into one by using rational exponents. We have the expression \( \sqrt[3]{5} \cdot \sqrt{2} \). The first radical is a cube root, and the second radical is a square root.

Step 2: Convert Radicals to Rational Exponents

Convert \( \sqrt[3]{5} \) into \( 5^{1/3} \) and \( \sqrt{2} \) into \( 2^{1/2} \) by expressing each radical as a power with a rational exponent.

Step 3: Combine the Expressions

When multiplying terms with the same base and exponents, you add the exponents. Since we are multiplying different bases, we keep them separate \[ 5^{1/3} \cdot 2^{1/2} \].

Step 4: Express as a Single Radical

Identify a common denominator for the exponents. The least common denominator of 3 and 2 is 6. Write 1/3 as 2/6, and 1/2 as 3/6. Now, express the entire expression using the common denominator:\[ 5^{2/6} \cdot 2^{3/6} = (5^2 \cdot 2^3)^{1/6} \]

Step 5: Simplify and Conclude

Calculate \( 5^2 = 25 \) and \( 2^3 = 8 \). Thus the expression becomes:\[ (25 \cdot 8)^{1/6} = 200^{1/6} \].The single radical expression combining \( \sqrt[3]{5} \cdot \sqrt{2} \) is \( \sqrt[6]{200} \).

Key Concepts

Understanding Radical Expressions

Radical expressions involve roots, like square roots, cube roots, and so on. These expressions are written using the radical sign, \( \sqrt{} \), or sometimes with a specific index for other roots, like \( \sqrt[3]{} \) for cube roots. Radicals express the inverse operation of raising a number to a power. For example, the square root of a number is essentially asking, "What number, when multiplied by itself, results in this number?".

To work with radical expressions like \( \sqrt[3]{5} \cdot \sqrt{2} \), you are often required to manipulate them into a more manageable form, such as a single radical expression or a simplification using rational exponents. This adjustment facilitates easier calculations and a more standard form for further mathematical operations.

Delving Into Rational Exponents

Rational exponents provide an alternative way to express roots, offering a more unified way to handle powers and roots simultaneously. Instead of using \( \sqrt[3]{5} \), you can express this using a rational exponent: \( 5^{1/3} \). Here, the denominator of the exponent indicates the root's degree.

Similarly, \( \sqrt{2} \) is expressed as \( 2^{1/2} \). The concept of rational exponents makes it easier to apply arithmetic operations like multiplication, which might otherwise be cumbersome with radicals alone. This transformation allows for a straightforward application of the properties of exponents, facilitating the combination and simplification of expressions.

Effective Simplification of Expressions

Simplification of expressions is a process of finding an equivalent expression that is simpler or more elegant. When dealing with expressions like \( 5^{1/3} \cdot 2^{1/2} \), we can combine these powers by managing their exponents.

The goal is to convert different-root expressions into a harmonized form using a common denominator for the exponents, which makes it possible to combine or further simplify them.

For instance, by finding a common denominator for \( 1/3 \) and \( 1/2 \) (which is 6), we're able to rewrite \( 5^{1/3} \) as \( 5^{2/6} \) and \( 2^{1/2} \) as \( 2^{3/6} \). This adjustment allows the expression to be combined and rewritten as \( (5^2 \cdot 2^3)^{1/6} \), simplifying the calculation process.

The Importance of LCM of Denominators

The Least Common Multiple (LCM) of denominators helps in adding, subtracting, or, as in this context, comparing and combining expressions with rational exponents. In our problem, converting different roots to a uniform base involves determining the LCM.

For the denominators 3 and 2 present in our expression \( \sqrt[3]{5} \cdot \sqrt{2} \), the smallest number they both divide into completely is 6. This discovery guides us to express the exponents in 6ths, i.e., \( 1/3 \) as \( 2/6 \) and \( 1/2 \) as \( 3/6 \).

Using the LCM in this manner ensures that all parts of the expression can be elegantly written under a common power, simplifying it ultimately to \( \sqrt[6]{200} \). This approach not only makes the process more streamlined but also ensures clarity and accuracy in solving such problems.