Problem 5 - Algebra A Combined Function

Question

Use the product rule to multiply. Assume that all variables represent positive real numbers. $$ \sqrt[3]{4} \cdot \sqrt[3]{9} $$

Step-by-Step Solution

Answer

\(\sqrt[3]{36}\)

Step 1: Identify the Problem

We need to multiply two cube roots: \(\sqrt[3]{4}\) and \(\sqrt[3]{9}\). The product rule states that the product of two like terms with roots can be written as a single root of the product of the terms. Therefore, we can apply the product rule to solve this.

Step 2: Apply the Product Rule

According to the product rule for roots, \(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}\). Here, \(n = 3\), \(a = 4\), and \(b = 9\). So, we can rewrite the expression as \(\sqrt[3]{4 \cdot 9}\).

Step 3: Multiply the Numbers Inside the Cube Root

Calculate the product \(4 \cdot 9\), which equals \(36\). Therefore, our expression simplifies to \(\sqrt[3]{36}\).

Step 4: Simplify the Cube Root, if Possible

Check if 36 can be simplified further into a perfect cube. Since 36 is not a perfect cube, \(\sqrt[3]{36}\) is the simplest form possible for this expression.

Key Concepts

Cube Roots

A cube root is a special type of radical that involves a number that must be multiplied by itself twice to equal another number. Essentially, it is the number that, when multiplied by itself three times, gives the original value under the radical. For example, the cube root of 8 is 2, because when you multiply 2 by itself twice more (2 \( \times \) 2 \( \times \) 2), you get 8.
Understanding cube roots allows you to work with equations and expressions where variables or numbers are raised to the power of 3 and need to be simplified back down. This is particularly useful in algebra when solving cube root equations or simplifying expressions with cube roots.
It's important to remember:

• The cube root is represented by the radical sign with a small three (\( \sqrt[3]{} \)).
• The cube root of a non-perfect cube often remains messy, as not all numbers have a neat cube root.
• Cube roots can apply to positive and negative numbers, unlike square roots which are often limited to non-negative numbers in basic algebra.

Multiplying Radicals

Multiplying radicals, such as cube roots, involves using the product rule for radicals. This rule helps to simplify the multiplication of two similar radicals. When you have the same index for both radicals (such as both being cube roots), you can combine them under one radical by multiplying their radicands (the numbers inside the radical).
Consider the product rule:

• If you have \( \sqrt[n]{a} \cdot \sqrt[n]{b} \), you can write it as \( \sqrt[n]{a \cdot b} \).
• This only works when the numbers or expressions under the roots (radicands) are positive and the roots are the same type (e.g., both cube roots).

For example, multiplying \( \sqrt[3]{4} \) by \( \sqrt[3]{9} \) uses this rule efficiently. You multiply the numbers inside the radicals together, resulting in one cube root: \( \sqrt[3]{4 \cdot 9} = \sqrt[3]{36} \). This combines the two separate radicals into one, simplifying calculations and expressions.

Simplifying Expressions

Simplifying expressions involving radicals can often make an equation easier to understand and work with. After applying rules like the product rule, the next step is simplifying the expression as much as possible by looking for perfect powers, factorization, or any common factors.
When you computed the radical \( \sqrt[3]{36} \), simplifying it might involve checking if 36 is a perfect cube which it is not. Therefore, \( \sqrt[3]{36} \) is in its simplest form in this context, as there are no "clean" cube roots to extract.
In practice, simplifying expressions might also involve:

• Reducing fractions under a radical sign before multiplying or dividing.
• Breaking down numbers to see if any components can be extracted from the radical into a simpler form.
• Combining like terms if multiple radicals are involved in an expression to make calculation easier and straightforward.

Simplifying expressions allows for neater presentation and often reveals a clearer path to the solution for algebraic problems.