Problem 38 - Algebra 2

Question

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt[3]{4} \cdot \sqrt[3]{80} $$

Step-by-Step Solution

Answer

The final answer is \(\sqrt[3]{320}\).

Step 1: Understand the Rules of Multiplying Radicals

The rule states that the product of \(\sqrt[n]{a}\) and \(\sqrt[n]{b}\) is equal to \(\sqrt[n]{a \cdot b}\) when 'n' is the same for both radicals. In this case, we have two cubic roots, so we can directly multiply the numbers under the roots.

Step 2: Implement the Rule

Applying the rule to our problem, we multiply the numbers under the cubic root. So it becomes like this \(\sqrt[3]{4} \cdot \sqrt[3]{80}\) equals to \(\sqrt[3]{4 \cdot 80}\).

Step 3: Calculate

Simplify \(\sqrt[3]{4 \cdot 80}\) to \(\sqrt[3]{320}\).

Key Concepts

Multiplying Radicals

When you multiply radicals, it's simple if you remember a key rule: you can multiply the numbers inside the same type of radical. For example, if you have two cubic roots, like in the exercise \( \sqrt[3]{4} \cdot \sqrt[3]{80} \), follow these steps:

• Ensure the radicals are of the same type — in this case, cubic roots.
• Multiply the numbers under the cube root signs: \( 4 \times 80 \).
• Combine them under one single radical sign: \( \sqrt[3]{320} \).

This approach is beneficial because it simplifies the expression before any further calculations. Remember that this method only works if the roots match; you can't multiply a square root with a cubic root. Understanding and correctly applying this rule streamlines the entire simplifying process.

Cubic Roots

Cubic roots, often represented as \( \sqrt[3]{x} \) or written with a small 3 above the root symbol, are a special type of radical. They tell us the number that, when used three times in multiplication, gives the original number. Let's illustrate this with \( \sqrt[3]{8} = 2 \), since \( 2 \times 2 \times 2 = 8 \).
Understanding cubic roots is crucial for simplifying expressions, as seen in our exercise of multiplying \( \sqrt[3]{4} \) and \( \sqrt[3]{80} \). Here's how cubic roots help us:

• They provide a way to reverse cubic exponentiation, which is multiplying three times.
• They allow us to represent numbers in a form that can be easily manipulated, especially in radical expressions.
• Comprehending how to multiply and simplify them aids in rationalizing denominators.

Recognizing and working with cubic roots can help unlock more complex mathematical problems, making them a powerful tool in your math toolkit.

Simplifying Expressions

Simplifying expressions is crucial for clarity and ease of calculation. It's all about breaking down a complex problem into simpler parts. Let's use the expression \( \sqrt[3]{320} \) from our previous multiplication steps. Simplification is achieved by:

• Factoring the number inside the radical: \( 320 = 2 \times 2 \times 2 \times 2 \times 2 \times 5 \).
• Team up the triplet of 2's: \( 2^3 \), which equals 8.
• Separate these out as a standalone factor: \( \sqrt[3]{8} \times \sqrt[3]{40} \).
• Simplify \( \sqrt[3]{8} \) to 2 and keep \( \sqrt[3]{40} \).

Thus our expression becomes \( 2 \cdot \sqrt[3]{40} \). The goal is a simpler form, often leading to a more easily understandable solution. Mastering these steps increases mathematical efficiency and reduces errors, making it easier to tackle more complex problems in the future.