Problem 34 - Algebra 2 and Trigonometry

Question

In \(11-38,\) evaluate each expression in the set of real numbers. $$ \sqrt[3]{\frac{1,000}{a^{3}},} a \neq 0 $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{10}{a} \).

Step 1 - Simplify the expression inside the cube root

The given expression is \( \sqrt[3]{\frac{1,000}{a^3}} \). First, simplify the fraction inside the cube root: \( \frac{1,000}{a^3} \) is equivalent to \( \frac{10^3}{a^3} \).

Step 2 - Apply the cube root to the fraction

Use the property of cube roots that states \( \sqrt[3]{\frac{b}{c}} = \frac{\sqrt[3]{b}}{\sqrt[3]{c}} \) to separate the cube root: \( \sqrt[3]{\frac{10^3}{a^3}} = \frac{\sqrt[3]{10^3}}{\sqrt[3]{a^3}} \).

Step 3 - Evaluate the cube roots separately

Find the cube root of the numerator: \( \sqrt[3]{10^3} = 10 \), because \( 10^3 = 1,000 \). For the denominator, \( \sqrt[3]{a^3} = a \).

Step 4 - Write the final simplified expression

Combine the results from Step 3 to get \( \frac{10}{a} \). Therefore, the simplified expression is \( \frac{10}{a} \).

Key Concepts

Real Numbers

Real numbers are a critical set of numbers used in mathematics, including both rational and irrational numbers. This group consists of everything from whole numbers to fractions and decimals, essentially encompassing any number you can place on an endless number line.
Real numbers include:

• Whole numbers, like 0, 1, 2, 3, and so on.
• Integers, which are positive and negative whole numbers.
• Rational numbers, like fractions or repeating decimals, such as \(\frac{1}{2}\) or 0.333....
• Irrational numbers, which cannot be expressed as a simple fraction, like \(\sqrt{2}\) or \(\pi\).

When dealing with cube roots, it's important to understand that they belong to the set of real numbers as well. For example, the cube root of a positive number is also a positive real number. In our exercise, \( \sqrt[3]{10^3} = 10 \) is a real number because 10 is a part of the set of rational whole numbers. Even though cube roots may seem complex, they often bring us back to a simple, manageable real number.

Simplify Expression

Simplifying an expression involves breaking it down to its simplest form without altering its original value. This is crucial in mathematics because it makes expressions easier to understand and work with. When simplifying expressions involving cube roots, it's helpful to separate components and deal with them individually. In our exercise with \( \sqrt[3]{\frac{10^3}{a^3}} \), we first simplified the inner fraction as \( \frac{10^3}{a^3} \). Breaking this down further by applying cube roots individually, we separated the numerator from the denominator.Using properties of cube roots, such as \( \sqrt[3]{b} \times \sqrt[3]{c} = \sqrt[3]{bc} \) and \( \sqrt[3]{\frac{b}{c}} = \frac{\sqrt[3]{b}}{\sqrt[3]{c}} \), helps to clearly and effectively simplify expressions. Ultimately, this process led us to a cleaner, simplified result of \( \frac{10}{a} \), free of radicals and much easier to interpret.

Evaluate Expression

To evaluate an expression means to calculate its value. This involves applying mathematical operations to reach a final, tangible numerical solution. In the original exercise, we evaluated an expression involving cube roots by first simplifying it. By understanding that \( \sqrt[3]{10^3} = 10 \) because \( 10^3 = 1,000 \), we could easily evaluate the cube root of the numerator. Similarly, applying the cube root to the denominator, \( a^3 \), gives us \( a \), since cube roots and cube powers cancel each other out.Thus, evaluating \( \sqrt[3]{\frac{10^3}{a^3}} \) simply involved using our basic understanding of math properties to determine that the expression simplifies to \( \frac{10}{a} \). This final result is much simpler to interpret and work with, demonstrating the power of evaluation methods in simplifying and solving mathematical problems effectively.