Problem 21 - Elementary and Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ \sqrt[3]{2} \cdot \sqrt[3]{12} $$

Step-by-Step Solution

Answer

The product simplifies to \(2 \sqrt[3]{3}\).

Step 1: Express Radicals as Powers

First, we need to express the cube roots using exponent notation. We can write the cube root of a number as that number raised to the power of \( \frac{1}{3} \). Thus, \( \sqrt[3]{2} = 2^{\frac{1}{3}} \) and \( \sqrt[3]{12} = 12^{\frac{1}{3}} \).

Step 2: Apply the Product of Powers Property

Using the property of exponents that states \(a^{m} \cdot a^{n} = a^{m+n}\), we multiply the expressions: \(2^{\frac{1}{3}} \cdot 12^{\frac{1}{3}} = (2 \cdot 12)^{\frac{1}{3}} \).

Step 3: Simplify Inside the Radical

Multiply the numbers inside the parentheses: \(2 \cdot 12 = 24\). So we have \((2 \cdot 12)^{\frac{1}{3}} = 24^{\frac{1}{3}}\).

Step 4: Breakdown 24 to Simplify

Factor 24 to simplify further: \(24 = 2^3 \cdot 3\). So, \(24^{\frac{1}{3}} = (2^3 \cdot 3)^{\frac{1}{3}}\).

Step 5: Apply the Power of a Product Property

Applying the property \((a \cdot b)^m = a^m \cdot b^m\), we have: \((2^3 \cdot 3)^{\frac{1}{3}} = (2^3)^{\frac{1}{3}} \cdot 3^{\frac{1}{3}} = 2^{\frac{3}{3}} \cdot 3^{\frac{1}{3}}\).

Step 6: Simplify the Expression

Since \(2^{\frac{3}{3}} = 2^1 = 2\), we can simplify the expression to \(2 \cdot 3^{\frac{1}{3}}\).

Step 7: Write the Final Simplified Form

Thus, the simplified form of the expression is \(2 \sqrt[3]{3}\).

Key Concepts

Product of Powers

To understand the concept of the "product of powers," we begin by expressing radicals as powers. When we have cube roots, these can be rewritten as numbers raised to the power of \(\frac{1}{3}\). For example, \(\sqrt[3]{2}\) is equivalent to \(2^{\frac{1}{3}}\). This allows us to correctly apply properties of exponents. The product of powers property states that when multiplying like bases with exponents, you add the exponents together. Mathematically, \[a^m \cdot a^n = a^{m+n}\] Here, since the bases are numbers, \(2^{\frac{1}{3}} \cdot 12^{\frac{1}{3}}\) isn't directly combining because the bases are different (2 and 12). However, understanding powers equips us to rearrange and manipulate expressions, including combining like bases. If the bases were the same, we would simply add \(m\) and \(n\) together. By rewriting and combining, such as multiplying under a single radical (when possible), we leverage this powerful tool to simplify expressions effectively.

Simplifying Radicals

Simplifying radicals involves breaking down complex root expressions into simpler, more manageable forms. When we encounter a problem like \(\sqrt[3]{24}\), our task is to simplify the expression by finding perfect cubes inside the radical, if any.

• First, factor the number into its prime components: for instance, 24 factors to \(2^3 \cdot 3\).
• Next, identify a perfect cube factor. Here, \(2^3\) is a perfect cube of 2.

This means \((2^3 \cdot 3)^{\frac{1}{3}}\) can be simplified because the cube root of \(2^3\) is 2.Thus, simplifying \(24^{\frac{1}{3}}\) results in \(2 \cdot 3^{\frac{1}{3}}\). Breaking down such expressions helps us write the result in a more digestible form, often with fewer radical components, making calculations easier.

Power of a Product

When dealing with expressions like \((a \cdot b)^m\), we use the "power of a product" property to streamline simplification. This property says that you can distribute the exponent \(m\) across the base components in a multiplication, such that: \[(a \cdot b)^m = a^m \cdot b^m\] Applying this concept can significantly clarify problems that include multiple terms under a single radical or exponent. For example, consider \((2^3 \cdot 3)^{\frac{1}{3}}\). By using this property:

• Distribute \(\frac{1}{3}\) to both 2 and 3 separately: \((2^3)^{\frac{1}{3}}\) and \(3^{\frac{1}{3}}\).
• This becomes \(2^{\frac{3}{3}} \cdot 3^{\frac{1}{3}}\).

The term \(2^{\frac{3}{3}}\) simplifies neatly to 2, illustrating the strength of the power of a product rule in dredging out complexity from an expression, revealing a more succinct form.