Problem 20 - Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ (2 \sqrt[3]{16})(-\sqrt[3]{4}) $$

Step-by-Step Solution

Step 1: Analyze the Expression

We are tasked with multiplying two cube roots: \((2 \sqrt[3]{16})(-\sqrt[3]{4})\). Our goal is to simplify this expression.

Step 2: Rewrite Cube Roots as Powers

Recall that a cube root of a number is equivalent to raising that number to the power of \(\frac{1}{3}\). Rewrite \(\sqrt[3]{16}\) as \(16^{\frac{1}{3}}\) and \(\sqrt[3]{4}\) as \(4^{\frac{1}{3}}\). The expression becomes \((2 \times 16^{\frac{1}{3}})(-1 \times 4^{\frac{1}{3}})\).

Key Concepts

Exponentiation

Exponentiation is the mathematical operation involving powers, where a number, known as the base, is raised to an exponent. In this context, an exponent refers to how many times the base number is multiplied by itself. For example, in the expression \(16^{\frac{1}{3}}\), the base is 16, and the exponent is \(\frac{1}{3}\), indicating the cube root of 16.

This is important when working with cube roots because it allows us to express roots using exponents. For cube roots, specifically, raising a number to the power of \(\frac{1}{3}\) gives us the equivalent of taking the cube root.

Understanding this relationship helps in simplifying expressions involving roots, as it allows for easier manipulation and multiplication of these terms by treating them as powers.

Simplifying Expressions

Simplifying expressions makes them easier to understand and evaluate. In our exercise, we simplified the expression by converting cube roots into powers. This makes mathematical operations more straightforward.

Initially, we had \((2 \sqrt[3]{16})(-\sqrt[3]{4})\). By expressing cube roots as powers: \(16^{\frac{1}{3}}\) and \(4^{\frac{1}{3}}\), the expression becomes \((2 \times 16^{\frac{1}{3}})\times (-1) \times 4^{\frac{1}{3}}\).

Breaking this into components, we then multiply the coefficients (2 and -1) and use properties of exponents to handle the radical terms:

• Combine the like exponents: \( 16^{\frac{1}{3}} \) and \( 4^{\frac{1}{3}} \)
• Simplify \( (2)(-1) = -2 \)

This gives a simplified form that is more compact and easier to calculate once fully resolved. The key is knowing how to manipulate exponents and maintain the integrity of the expression.

Positive Real Numbers

In mathematics, positive real numbers are numbers greater than zero, and they include all non-negative rational and irrational numbers. These numbers form the basis of our exercise here.

Understanding positive real numbers is crucial because all variables in our expression are restricted to being positive real numbers. This assumption allows us to simplify expressions without additional constraints related to sign changes from negative values.

When you work with cube roots or any radicals involving positive real numbers, remember these numbers retain their positivity after operations such as exponentiation.
This means that any real positive input under these transformations (like cube roots) will result in a valid, simplified real number output, maintaining consistent results.