Problem 28 - College Algebra with Modeling and Visualization

Question

If possible, simplify the expression by hand. If you cannot, approximate the answer to the nearest hundredth. Variables represent any real number. $$ \sqrt[3]{64} $$

Step-by-Step Solution

Answer

The cube root of 64 is 4.

Step 1: Identify the Expression

The expression given is \( \sqrt[3]{64} \). This indicates that we are looking for the cube root of 64.

Step 2: Determine the Cube Root

To determine the cube root of 64, we need to find a number which, when multiplied by itself three times, equals 64. Mathematically, we are seeking a number \( x \) such that \( x^3 = 64 \).

Step 3: Calculate the Result

By trial and error or knowledge of basic cubes, we know that \( 4 \times 4 \times 4 = 64 \). Therefore, \( \sqrt[3]{64} = 4 \).

Step 4: Verify the Solution

We verify by recalculating: \( 4 \times 4 = 16 \) and \( 16 \times 4 = 64 \), so indeed \( 4^3 = 64 \). Hence, \( \sqrt[3]{64} = 4 \).

Key Concepts

Cube Roots

Cube roots are numbers that we raise to the power of three to obtain a given number. In simpler terms, to find a cube root of a number like 64, we look for a number that, when multiplied by itself twice more, yields 64. It's a fundamental concept in algebra that helps us solve various mathematical problems.
Cube roots can be applied to real numbers and are especially useful when managing equations that involve cubics. Unlike square roots, every real number, whether positive or negative, has a cube root. This is because multiplying a negative number three times results in a negative number. Here's an important note:

• If the cube is a positive number, such as 64, the cube root will also be positive.
• If the cube is a negative number, like -64, the cube root will be negative.

To find cube roots without memorization, practice decomposing numbers into their prime factors or rely on the fundamental properties and small cube numbers like 1, 8, 27, 64, and so on.

Radical Expressions

Radical expressions are numbers that include roots. They are typically represented with a radical sign, such as square roots, cube roots, or even higher roots. When dealing with radical expressions, particularly cube roots, it's important to recognize the index of the radical.

• For example, in \( \sqrt[3]{64} \), the small 3 signifies it's a cube root rather than a square root, which usually doesn't display an index.
• Radicals allow us to express powers and roots in a concise way.

Working with these expressions requires a good understanding of how to simplify or manipulate them. Simplifying radicals might involve factoring the number under the root or identifying perfect powers. Understanding radical expressions is crucial as they form the basis for solving algebraic equations and evaluating expressions where roots are not easily removed or simplified.

Real Numbers

Real numbers encompass all the numbers we can find on the infinite number line. This includes both rational numbers (fractions, finite decimals) and irrational numbers (like \( \pi \) and \( \sqrt{2} \)). Real numbers are vital in algebra and everyday applications because they cover a vast range, including negative numbers, zero, and positive numbers.
When working with cube roots or other radicals, real numbers offer us the assurance that a cube root of a positive or negative number will always yield a real number as a solution. Here are a few important points:

• Every real number has a unique cube root.
• Real numbers are closed under addition, subtraction, multiplication, and division (except by zero).
• This closure property means operations between real numbers always result in real numbers.

Understanding real numbers helps us grasp the entirety of solutions available in algebra and supports our exploration of more complex number systems used in advanced mathematics.