Problem 69 - Algebra 2

Question

Simplify each expression. \(3^{\frac{1}{3}} \cdot 18^{\frac{1}{3}}\)

Step-by-Step Solution

Answer

The simplified expression is \(3 \cdot 2^{\frac{1}{3}}\).

Step 1: Apply the Property of Exponents

The property of exponents states that when you multiply terms with the same bases, you add the exponents. Therefore, \(3^{\frac{1}{3}} \cdot 18^{\frac{1}{3}}\) can be rewritten as \(3^{\frac{1}{3}} \cdot (3^2 \cdot 2)^{\frac{1}{3}}\), where 18 is expressed as a product of prime numbers.

Step 2: Simplify Expression

Using the properties of exponents again, the expression simplifies to \(3^{\frac{1}{3}} \cdot 3^{2 * \frac{1}{3}} \cdot 2^{\frac{1}{3}}\). This further simplifies to \(3^1 \cdot 2^{\frac{1}{3}}\), or \(3 \cdot 2^{\frac{1}{3}}\)

Key Concepts

Simplifying Expressions

When simplifying expressions, the goal is to rewrite the mathematical expression in its simplest form. This often involves performing operations and applying mathematical properties to make the expression easier to understand or evaluate. In this exercise, we simplified the expression \(3^{\frac{1}{3}} \cdot 18^{\frac{1}{3}}\) by utilizing exponent properties.

• First, we noticed that both terms in the expression were involving powers, which allowed us to use the properties of exponents.
• Specifically, when multiplying terms with exponents and the same base, you add the exponents together.

To further simplify, we used these exponent properties to rewrite and reduce the expression. This process showed that the base numbers could become the same, which helped us simplify the multiplication into a single power.

Fractional Exponents

Fractional exponents can initially seem confusing, but they are simply another way to represent roots. The numerator of a fractional exponent indicates the power, while the denominator indicates the root. Thus, \(a^{\frac{1}{3}}\) is the cube root of \(a\).

• In the expression \(3^{\frac{1}{3}} \cdot 18^{\frac{1}{3}}\), the \(\frac{1}{3}\) exponent signifies we are finding the cube root of both 3 and 18.
• By expressing 18 as \(3^2 \cdot 2\), it becomes easier to apply the fractional exponent to prime constituents.

Understanding the representation of roots in fractional exponent form simplifies the process of transforming complex expressions into more manageable terms. This is particularly useful in reductive operations like multiplying and dividing numbers with fractional exponents.

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks: prime numbers. Any number can be expressed as a product of prime factors. For example, 18 can be factored into \(2 \cdot 3^2\).

• This approach is useful in simplifying expressions, such as \(3^{\frac{1}{3}} \cdot 18^{\frac{1}{3}}\), because it allows us to rewrite terms in a form where exponent properties are more easily applied.
• By breaking down 18 into prime factors, we can address each factor's exponent independently, using the rules of exponents to simplify further.

Prime factorization can reveal hidden connections between terms that enable more straightforward calculations, especially when dealing with expressions that involve roots or fractional exponents.