Problem 34 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt[3]{3 z^{2}} \cdot \sqrt[3]{7 z} $$

Step-by-Step Solution

Answer

The simplified expression is \( z \sqrt[3]{21} \).

Step 1: Express as a Single Cube Root

The expression \( \sqrt[3]{3z^2} \cdot \sqrt[3]{7z} \) can be rewritten as \( \sqrt[3]{3z^2 \times 7z} \). We do this by using the property that \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} \). This simplifies the expression under a single cube root.

Step 2: Multiply Inside the Cube Root

Now multiply the terms inside the cube root: \[ 3z^2 \times 7z = 21z^3. \]This combines the coefficients and the powers of \( z \) to create \( 21z^3 \).

Step 3: Simplify the Cube Root

Extract the cube root of \( 21z^3 \): \[ \sqrt[3]{21z^3} = \sqrt[3]{21} \times z. \]This is because \( z^3 \) is a perfect cube, so its cube root simplifies to \( z \). Thus, we cannot further simplify \( \sqrt[3]{21} \) as it is not a perfect cube.

Key Concepts

Cube Roots

Understanding cube roots is crucial when simplifying radical expressions like the one in our example. The cube root of a number is a value that, when multiplied by itself three times, equals the original number. For instance, the cube root of 8 is 2, because

• \( 2 \times 2 \times 2 = 8 \).

Cube roots are denoted by the radical sign with a small three, called the index, positioned above the root symbol,

• \( \sqrt[3]{x} \).

This notation signifies a cube root operation. When dealing with expressions featuring cube roots, keep in mind that multiplication of cube roots can be combined under one cube root, as seen in the step-by-step solution above. This involves multiplying the expressions under the roots before taking the cube root of the entire product. It's important to simplify these expressions as much as possible, taking care to extract any perfect cubes for easy computation.

Properties of Exponents

Knowing the properties of exponents is vital in simplifying expressions with radicals, including cube roots. Exponents are numbers that denote how many times a base is multiplied by itself. For example, in the expression \( z^3 \), you multiply \( z \) with itself three times. Here are key properties that help in simplification:

• Product of Powers: \( a^m \times a^n = a^{m+n} \)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)
• Power of a Product: \((ab)^n = a^n \cdot b^n\)

In our original expression, the exponents helped combine terms under a single radical. By computing the total power of \( z \) in both terms, they were added: \( z^2 \times z = z^{2+1} = z^3 \). This allowed the simplification inside the cube root. Becoming comfortable with these properties is essential for effectively managing algebraic simplifications.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations. Simplifying algebraic expressions involves combining like terms and applying arithmetic operations as well as the properties of exponents and roots. In our problem, starting with expressions like \( 3z^2 \) and \( 7z \), conversion into a single radical was achieved by leveraging properties such as combining within a single cube root.

• Multiplication of algebraic terms: Combine coefficients (numerical values) together, and perform operations on variables using the properties of exponents.
• Simplification: Reduce the expression by extracting roots of perfect powers and combining terms effectively.

In this exercise, the process boiled down to simplifying to \( 21z^3 \) before taking final steps of obtaining \( z \) from the cube root \( \sqrt[3]{z^3} \). Understanding how to handle algebraic expressions, especially when they involve roots and exponents, is an invaluable skill in mathematics.