Problem 52 - College Algebra

Question

Evaluate each expression in Exercises \(49-60\), or indicate that the root is not a real number. $$\sqrt[3]{-125}$$

Step-by-Step Solution

Answer

The cube root of -125 is -5.

Step 1: Identify the Cube Root

The exercise provided is looking for the cube root of -125, which can be written as \( \sqrt[3]{-125} \).

Step 2: Apply the Cube Root

The cube root of -125, \(\sqrt[3]{-125}\), is the number which, when cubed (\(x^3\)), gives -125.

Step 3: Solve for Cube Root

For the problem \(\sqrt[3]{-125}\), the cube root is -5. This is because \((-5)^3\) equals to -125.

Key Concepts

Real Numbers

Real numbers are everywhere around us. They include every number we can think of, from the ones we use in our daily lives to the more complex ones found in mathematics. They span a range of numbers including:

• Whole numbers like 0, 1, and 10.
• Fractions like \(\frac{1}{2}\) or \(0.25\).
• Integers such as -3, 0, and 7.
• Irrational numbers like \(\pi\) (Pi) and \(\sqrt{2}\).
• Positive and negative numbers.

Because real numbers cover both the positive and negative sides, they include numbers such as -5 and 125, which you often find in cube root calculations. These numbers help in solving real-world problems and in tasks like our given problem of finding the cube root of -125. They provide a foundation for understanding concepts like cube roots, which ask for a real number that can be multiplied by itself three times to return the original value.

Negative Numbers

Negative numbers are unique because they are less than zero. We see them in various situations, like when temperatures drop below freezing or in financial contexts with debt. They are essential for calculations like finding cube roots, especially for negative results.When dealing with cube roots of negative numbers, the result is surprisingly simple. The cube root of a negative number will also be negative. For example, when finding the cube root of -125, you look for a number which, when multiplied by itself three times, results in -125. That number is -5 because \((-5)\times(-5)\times(-5)\) gives -125.It's interesting to note that cube roots are different from square roots in this respect. A square root of a negative number is not a real number, but cube roots always work out neatly, which is part of what makes them interesting to calculate.

Mathematical Expressions

Mathematical expressions are like a special language. They use numbers, symbols, and operators to convey mathematical ideas. Think of them as short-hand instructions that describe a calculation or a concept quickly and efficiently.In the exercise \(\sqrt[3]{-125}\), the expression tells us to find a number that, when cubed, equals -125. The symbol \(\sqrt[3]{}\) stands for the cube root, while the number inside the symbol is the value we're interested in.Breaking down mathematical expressions into steps makes them easier to tackle. For example:

• Recognize the operation: Look for symbols like \(\sqrt[]{}\) to identify that root calculation is needed.
• Understand the number: Know what the number represents and how it interacts with the operation.
• Solve step-by-step: Simplify and calculate piece-by-piece, as shown with breaking down \((-5)\times(-5)\times(-5)\).

By mastering these expression-processing skills, you can more easily solve complex problems and understand the principles behind them.