Problem 29 - Algebra 2 and Trigonometry

Question

In \(11-38,\) evaluate each expression in the set of real numbers. $$ \sqrt[3]{0.001} $$

Step-by-Step Solution

Answer

The value of \(\sqrt[3]{0.001}\) is \(0.1\).

Step 1: Identify the Expression

The given expression is \(\sqrt[3]{0.001}\). This means we are tasked with finding the cube root of 0.001.

Step 2: Convert to an Exponential Form

We can express the problem in exponential terms: \(0.001 = 10^{-3}\). Therefore, the expression becomes \( (10^{-3})^{1/3} \).

Step 3: Apply the Power Rule for Exponents

According to the power rule \((a^m)^n = a^{m \times n}\), we can simplify \((10^{-3})^{1/3}\) to \(10^{-3 \times 1/3} = 10^{-1}\).

Step 4: Evaluate the Final Answer

Since \(10^{-1} = \frac{1}{10}\), the final evaluation of the expression \(\sqrt[3]{0.001}\) is \(0.1\).

Key Concepts

Understanding Exponents

Exponents are mathematical notations used to express repeated multiplication of the same number by itself. When you see a number written with a small superscript number, that's an exponent at work. For example, in the expression \(2^3\), the base is 2, and the exponent is 3. This means you multiply 2 by itself three times, resulting in \(2 \times 2 \times 2 = 8\).

Exponents follow several important rules which make them especially handy.

• They allow large numbers to be expressed concisely.
• They help simplify multiplication and division processes, especially when dealing with the same base.
• They can represent both whole numbers and fractions, lending flexibility to mathematical operations involving powers.

Exponents are integral not just in pure mathematics, but also in scientific calculations and various engineering fields, as they provide a systematic way to denote powers and roots.

Exploring Exponential Notation

Exponential notation is a powerful method for representing numbers, particularly large or small values, using powers of ten. This notation involves writing numbers in the form of \(a \times 10^n\), where "a" is a coefficient that can be a fractional value, and "n" is the exponent that signifies the power of ten. For example, \(0.001\) can be expressed as \(1 \times 10^{-3}\), which makes calculations neat and easier to handle.

This notation is beneficial because it:

• Simplifies complex multiplications and divisions involving large numbers.
• Allows for straightforward conversion between numbers of varying scales, such as billions or thousands.
• Is frequently used in scientific fields to express measurements, such as distances in astronomy or sizes in cellular biology.

Learning to confidently express numbers in exponential notation forms the foundational skills required in higher education and professional scientific tasks.

Power Rule for Exponents Simplified

The power rule for exponents is fundamental in simplifying expressions where exponents are raised to another power. This rule states: \((a^m)^n = a^{m \times n}\). In simpler terms, when you have an exponent raised to another exponent, you multiply the two exponents together.

Using the power rule effectively allows for

• Condensing expressions involving repeated powers.
• Simplification of calculations involving cube roots, fourth roots, and beyond.
• Easing the process of algebraic manipulations in calculus and physics.

For instance, in the given exercise, the expression \((10^{-3})^{1/3}\) was simplified using the power rule to \(10^{-1}\). This transformation made evaluating the cube root of 0.001 much simpler and is illustrative of the utility of the power rule in practical problem-solving scenarios.