Problem 55 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive and write your answer in radical notation. $$ \sqrt{3} \cdot \sqrt[3]{3} $$

Step-by-Step Solution

Answer

The simplified expression is \( \sqrt[6]{3^5} \).

Step 1: Understand the Expression

The given expression is \( \sqrt{3} \cdot \sqrt[3]{3} \). This involves two different radicals: a square root and a cube root of the same base, 3.

Step 2: Express as Exponents

Express the radicals using exponents. The square root can be written as \( 3^{1/2} \) and the cube root as \( 3^{1/3} \). Thus, the expression becomes \( 3^{1/2} \times 3^{1/3} \).

Step 3: Apply Properties of Exponents

According to the properties of exponents, specifically the product of powers property, when multiplying terms with the same base, we can add their exponents. Therefore, \( 3^{1/2} \times 3^{1/3} = 3^{1/2 + 1/3} \).

Step 4: Find a Common Denominator

To add \( 1/2 \) and \( 1/3 \), a common denominator is needed. The least common denominator of 2 and 3 is 6. So, \( 1/2 = 3/6 \) and \( 1/3 = 2/6 \).

Step 5: Add the Exponents

Add the fractions: \( 3/6 + 2/6 = 5/6 \). Hence, the exponent becomes \( 3^{5/6} \).

Step 6: Write Answer in Radical Notation

Convert \( 3^{5/6} \) back into radical notation. The expression \( 3^{5/6} \) can be rewritten as \( \sqrt[6]{3^5} \).

Key Concepts

Properties of Exponents

Exponents play a vital role in simplifying expressions, especially when dealing with radicals. They represent how many times a number, called the base, is multiplied by itself. For example, in the expression \( 3^4 \), we multiply 3 by itself four times: \( 3 \times 3 \times 3 \times 3 = 81 \).

One important property of exponents is when multiplying two expressions with the same base, like \( a^m \times a^n \), we simply add the exponents: \( a^{m+n} \). This property is incredibly useful for simplification.

• For example, \( 3^{1/2} \times 3^{1/3} \) becomes \( 3^{1/2 + 1/3} \).
• We then add these fractions by finding a common denominator.

Another property is the power of a power, where \((a^m)^n = a^{m\times n} \). These properties help us move between radical notation and exponents seamlessly, making expressions simpler.

Remembering these properties aids in understanding and solving complex algebraic expressions.

Radical Notation

Radical notation provides a way to express roots in mathematics, allowing us to write expressions involving square roots, cube roots, and higher-order roots. The radical sign \( \sqrt{} \) indicates a root, and the number under the radical is called the radicand.

In our exercise, we dealt initially with \( \sqrt{3} \) and \( \sqrt[3]{3} \). Translating these into exponent form, we have \( 3^{1/2} \) and \( 3^{1/3} \) respectively.

• Square roots translate to the exponent \( 1/2 \).
• Cube roots translate to the exponent \( 1/3 \).

When simplifying, we often convert between radical and exponential notation. This makes it easier to apply exponent properties and streamline calculations.

In the final answer, \( 3^{5/6} \) returns to radical notation as \( \sqrt[6]{3^5} \). This shows both the order of the root and the power of the radicand, offering a complete and simplified view of the expression's value.

Simplification of Expressions

Simplification is the process of rewriting an expression in its simplest form, often involving reduction of complexity without changing the value. By using properties of exponents and converting between exponent and radical forms, we can simplify expressions significantly.

With algebraic expressions such as \( \sqrt{3} \cdot \sqrt[3]{3} \), simplification involves several steps:

• Express both radicals as exponents \( 3^{1/2} \) and \( 3^{1/3} \).
• Use properties of exponents to combine them: \( 3^{1/2 + 1/3} = 3^{5/6} \).
• Convert the exponent back into radical notation for the most simplified form: \( \sqrt[6]{3^5} \).

This transformation from complex to simple involves a deep understanding of the relationships between different forms and properties in mathematics.

Simplifying correctly ensures that expressions are easy to read and interpret while retaining mathematical accuracy, a skill essential for advancing in algebra and beyond.