Problem 6 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt[3]{x} \cdot \sqrt[3]{x^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( x \).

Step 1: Understand the Expression

The expression we need to simplify is \( \sqrt[3]{x} \cdot \sqrt[3]{x^{2}} \). This involves the multiplication of two cube roots.

Step 2: Apply the Radical Product Rule

When multiplying two roots of the same degree, you can multiply the radicands and keep them under a single root. Thus, \( \sqrt[3]{x} \cdot \sqrt[3]{x^{2}} = \sqrt[3]{x \cdot x^{2}} \).

Step 3: Simplify the Radicand

The expression inside the cube root becomes \( x \cdot x^{2} = x^{1 + 2} = x^{3} \). So, \( \sqrt[3]{x \cdot x^{2}} = \sqrt[3]{x^{3}} \).

Step 4: Simplify the Cube Root

Taking a cube root of something cubed simplifies directly to the original base: \( \sqrt[3]{x^{3}} = x \). This is because the cube and the cube root cancel each other out.

Key Concepts

Cube Roots

Cube roots can seem a bit tricky at first, but they're just another type of radical expression. Here's a simple way to think about cube roots: while a square root asks the question "what number multiplied by itself twice gives me this number?" a cube root is asking "what number multiplied by itself three times gives me this number?" For example, the cube root of 8 is 2, because multiplying 2 by itself three times gives 8 (i.e., 2 * 2 * 2 = 8). In notation, we'd express the cube root of 8 as \( \sqrt[3]{8} = 2 \). Cube roots are typically identified by the small little 3 in front of the radical sign.In algebra, when you come across cube roots combined with variables, like \( \sqrt[3]{x} \), it's handy to rewrite the expression using exponents. In this case, \( \sqrt[3]{x} \) is the same as \( x^{\frac{1}{3}} \). This change can make it easier to simplify expressions, especially when you're dealing with multiplication or division of cube roots.

Radical Expressions

A radical expression involves roots, which are indicated by the radical sign \( \sqrt{} \). These can include square roots, cube roots, or even higher roots. When dealing with radical expressions, it's important to remember a few rules to simplify them efficiently.

• Roots of the same degree can often be multiplied or divided easily. For instance, \( \sqrt[3]{x} \cdot \sqrt[3]{y} = \sqrt[3]{xy} \). The same degree means they share the same index, in this case, 3 for cube roots.
• Always ensure the radicands (the numbers or expressions inside the radical sign) are positive to prevent extra complications.
• Remember, you can simplify radicals by breaking them down into simpler components – similar to how you simplify fractions or exponents.

Radical expressions become much less daunting once you practice these straightforward rules. They allow you to seamlessly transition between different formats of expressions and solve problems more efficiently.

Exponent Rules

Exponent rules are essential when dealing with radical expressions since a lot of simplification requires converting radicals to exponents. Let's brush up on a few key rules:

• **Product of Powers Rule:** When you multiply like bases, you add the exponents: \( a^m \cdot a^n = a^{m+n} \).
• **Power of a Power Rule:** When raising a power to another power, you multiply the exponents: \( (a^m)^n = a^{m \cdot n} \).
• **Power of a Product Rule:** Raise each factor in the base to the exponent: \( (ab)^n = a^n \cdot b^n \).
• **Radicals and Fractional Exponents:** A root can be expressed as a fractional exponent, such as \( \sqrt[3]{x} = x^{\frac{1}{3}} \).

These rules simplify expressions elegantly and help unify our understanding of algebraic expressions. In the original exercise, we applied the product of powers rule to simplify \( x \cdot x^2 = x^{1+2} = x^3 \) and found that \( \sqrt[3]{x^3} = x \). This is because a radical application like cube root cancels out an exponent of three.