Problem 2 - Algebra 2

Question

Multiply, if possible. Then simplify. $$ \sqrt[3]{4} \cdot \sqrt[3]{16} $$

Step-by-Step Solution

Answer

The simplified value is \(4\).

Step 1: Multiply Numbers Under Cube Roots

The first step is to multiply the numbers under the cube roots. So, \( \sqrt[3]{4} \cdot \sqrt[3]{16} \) becomes \( \sqrt[3]{4 \cdot 16} \), which simplifies to \( \sqrt[3]{64} \).

Step 2: Simplify the Cube Root

The next step is to simplify the cube root. So we get the cube root of 64, which is \(4\), since \(4^3=64\).

Key Concepts

Simplifying Radicals

Simplifying radicals involves rewriting a radical expression in its simplest form. For cube roots, such as \( \sqrt[3]{64} \), we look for a number which, when multiplied by itself three times, gives 64.

• Identifying the Cube: When dealing with cube roots, it is natural to search for a perfect cube that equals the radicand (the number inside the root). For 64, the number 4 works because \( 4 \times 4 \times 4 = 64 \).
• Reduction Process: If the radicand is not a perfect cube, it’s often helpful to break it down into smaller factors, aiming to identify a perfect cube within those factors.

After finding that \( 4^3 = 64 \), the cube root simplifies directly to 4. Simplifying radicals is a key skill, as it allows you to understand and solve mathematical expressions more easily.

Multiplying Radicals

When you multiply radicals, such as cube roots, you combine their radicands (numbers inside the root) under a single radical sign. In the given problem, we multiply \( \sqrt[3]{4} \) and \( \sqrt[3]{16} \) to get \( \sqrt[3]{4 \cdot 16} \), which simplifies to \( \sqrt[3]{64} \).

• Same Root Index: The roots can be multiplied directly if they have the same index, or type of root, as in cube roots here. This means treating them as normal numbers once they are under the same radical sign.
• Combining Factors: It helps to multiply the factors of the radicands first before computing the final product if multiple terms are involved.

This process is crucial for simplifying expressions and solving equations that involve radicals. Always group and simplify carefully to maintain accuracy.

Properties of Exponents

The properties of exponents are important when simplifying expressions with radicals. Recall that roots can also be expressed using fractional exponents. For example, \( \sqrt[3]{x} \) is the same as \( x^{1/3} \).

• Exponential Rules: These rules allow us to transform roots into familiar exponent forms, enhancing flexibility in solving problems.
• Combining Powers: When multiplying expressions with the same base, we add their exponents. For example, \( x^{a} \times x^{b} = x^{a+b} \). This property supports simplifying and multiplying radicals, allowing you to use exponent rules to streamline calculations.

Such properties are fundamental to algebra, especially when you're tasked with working through radical and exponential expressions in various contexts. Understanding them can simplify not just individual steps, but the entire problem-solving process.