Problem 114 - Intermediate Algebra

Question

Answer true or false. Assume all radicals represent nonzero real numbers. $$ \frac{\sqrt[3]{12}}{\sqrt[3]{4}}=\sqrt[3]{8} $$

Step-by-Step Solution

Answer

False.

Step 1: Simplify the fraction

First, simplify the fraction \( \frac{\sqrt[3]{12}}{\sqrt[3]{4}} \). We use the property of roots: \( \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}} \). Therefore, \( \frac{\sqrt[3]{12}}{\sqrt[3]{4}} = \sqrt[3]{\frac{12}{4}} \).

Step 2: Simplify the fraction inside the cube root

Calculate \( \frac{12}{4} \) inside the cube root. The division results in 3. Thus, we have \( \frac{\sqrt[3]{12}}{\sqrt[3]{4}} = \sqrt[3]{3} \).

Step 3: Compare the expressions

We now compare \( \sqrt[3]{3} \) to \( \sqrt[3]{8} \). Since 3 is not equal to 8, \( \sqrt[3]{3} eq \sqrt[3]{8} \). Hence, the statement \( \frac{\sqrt[3]{12}}{\sqrt[3]{4}} = \sqrt[3]{8} \) is false.

Key Concepts

Cube Roots

Understanding cube roots is fundamental in working with radicals. The cube root of a number is a value that, when multiplied by itself three times, results in the original number. For example, the cube root of 8 is 2, because 2 multiplied by itself three times (2 x 2 x 2) equals 8. When we see the expression \( \sqrt[3]{a} \), it indicates the cube root of a number \( a \). It's similar to square roots but with an emphasis on cubes, meaning three factors of the same number.

Cube roots are particularly interesting because they offer insights into the volume and growth rates in geometry and algebra.

• \( \sqrt[3]{27} = 3 \) because \( 3 \times 3 \times 3 = 27 \)
• \( \sqrt[3]{1} = 1 \) since \( 1^3 = 1 \)
• Negative numbers also have real cube roots, so \( \sqrt[3]{-8} = -2 \) since \(-2 \times -2 \times -2 = -8\)

This property makes cube roots useful in solving various equations and understanding wider concepts in math.

Simplifying Fractions

Simplifying fractions, including those involving cube roots, is a crucial skill in algebra. The process involves reducing the fraction to its simplest form or making the expression easier to work with. In the context of radicals, simplifying a fraction often means applying a property of roots: \( \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}} \).

Let's look at this step-by-step:

• For the expression \( \frac{\sqrt[3]{12}}{\sqrt[3]{4}} \), we use the cube root property: \( \sqrt[3]{\frac{12}{4}} \).
• Perform the division inside the radical, \( \frac{12}{4} = 3 \).
• This simplifies the expression to \( \sqrt[3]{3} \).

Simplifying helps in comparing and solving expressions efficiently. It breaks down the work into manageable parts and reduces the potential for errors in calculation, especially when dealing with radicals or higher-order roots.

Real Numbers

Real numbers form the foundation of everyday arithmetic and algebra. They include a variety of numbers: integers, fractions, decimals, and irrational numbers. Essentially, any number that can be found on the number line is a real number. When working with cube roots and other radicals, we often assume they are dealing with nonzero real numbers.

Here’s what makes up the realm of real numbers:

• **Integers** include numbers like -3, 0, 1, and 2.
• **Fractions** and **decimals** such as \( \frac{1}{2} \), 0.75, etc., are also real.
• **Irrational numbers** include roots like \( \sqrt{2} \), which can’t be exactly expressed as a fraction.

The statement in the exercise that all radicals represent nonzero real numbers is significant because it implies that none of the terms can be zero. This ensures that operations like division are valid and that we can confidently use properties of roots without encountering undefined situations. Real numbers are versatile tools, used across mathematics, ensuring we can manipulate and understand various expressions and equations effortlessly.