Problem 72 - Elementary and Intermediate Algebra

Question

Simplify each cube root. See Example \(6 .\) $$ \sqrt[3]{8} $$

Step-by-Step Solution

Answer

The simplified cube root of 8 is 2.

Step 1: Understand the Cube Root

In this exercise, we want to simplify the cube root \( \sqrt[3]{8} \). A cube root of a number \( x \) is a number \( y \) such that \( y^3 = x \). Therefore, we need to find a number \( y \) such that when it is cubed, the result is 8.

Step 2: Identify the Perfect Cube

Consider the number 8. We need to determine if it is a perfect cube, i.e., if there is an integer that can be multiplied by itself twice to give 8. We know that \( 2^3 = 8 \). Thus, 8 is a perfect cube. This means that the cube root can be simplified to this integer.

Step 3: Simplify the Cube Root

Since we found that \( 2^3 = 8 \), the cube root \( \sqrt[3]{8} \) simplifies to \( 2 \). Therefore, the expression \( \sqrt[3]{8} \) can be simplified to \( 2 \).

Key Concepts

Perfect Cubes

A perfect cube is a number that can be expressed as the cube of an integer. If you take an integer and multiply it by itself twice, the resulting product is a perfect cube. To illustrate, when you take the number 2 and raise it to the power of 3, you multiply two by itself and then by itself again: \[ 2 \times 2 \times 2 = 8 \]This means that 8 is a perfect cube. Knowing whether a number is a perfect cube is essential when you're tasked with simplifying cube roots. This property helps you quickly determine the cube root of a number, as it is simply the original integer raised to the third power.

• Examples of perfect cubes include: 1, 8, 27, 64, and 125.
• These correspond to: \[1^3, 2^3, 3^3, 4^3, \text{and}\ 5^3 \text{ respectively.}\]

Identifying perfect cubes can simplify cube root calculations, making your math journey much smoother.

Cube Root Calculation

Calculating the cube root of a number is an important skill, especially when simplifying expressions. The cube root of a number \( x \) is a value \( y \) such that \( y^3 = x \). To find the cube root, you are essentially asking, "What number multiplied by itself twice will give this value?" When dealing with a perfect cube, the cube root calculation simplifies drastically. Like in our example, \( \sqrt[3]{8} \) is easy to compute, because we know that \( 2^3 = 8 \). So, the cube root of 8 is 2.

• For non-perfect cubes, calculating the cube root may not result in an integer.
• Instead, the result could be an irrational number.

Solving cube root problems involves recognizing the structure of numbers and understanding how powers of three play a part in calculations.

Integer Exponents

Understanding integer exponents is crucial for simplifying expressions involving cube roots and perfect cubes. An exponent represents how many times a number, known as the base, is multiplied by itself. Integer exponents are particularly common because they keep calculations straightforward, involving what we call whole number exponents. For cubing, this means raising a number to the power of 3, as in \( y^3 \). When \( y^3 \) equals a whole number, it signifies that number is a perfect cube. For instance, 2 raised to the third power, \( 2^3 \), equals 8.

• Exponents show how many times to multiply a base by itself.
• They simplify multiplication operations into easy-to-understand power terms.

Mastering integer exponents gives you a solid foundation for tackling more complex algebraic and mathematical challenges. In particular, they allow you to easily simplify cube roots, making the math much more approachable.