Problem 2 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$\sqrt[3]{125}$$

Step-by-Step Solution

Answer

The cube root of 125 is 5.

Step 1: Express the number as a cube of a number

The number \(125\) may be expressed as \(5^3\), because 5 * 5 * 5 is 125. This forms the perfect cube, and it is what we'll use to simplify the cube root.

Step 2: Apply the cube root property

The cube root of a cube is the original number. Therefore, if we know \(125 = 5^3\), then the cube root of \(125\) can be written as the cube root of \((5^3)\). And the cube root of \((5^3)\) is simply 5.

Key Concepts

Simplifying Expressions

Simplifying mathematical expressions is just like decluttering a room; it's about finding the most straightforward form of something that might initially look complex. In the context of cube roots, this often means transforming the number under the radical sign into a power of another number.

For instance, say we want to simplify \(\sqrt[3]{125}\). We first look for a number that, when multiplied by itself three times (or cubed), gives us 125. Recognizing that \(5 \times 5 \times 5 = 5^3 = 125\) is the key. By changing 125 into \(5^3\), we’ve turned an intimidating expression into something much friendlier.

In simplifying expressions, always look for patterns or relationships that convert complex-looking numbers into simpler, more recognizable forms. It’s like finding the right code to unlock a puzzle. For cube roots, especially, the goal is to express the radicand—that’s the number under the radical—as a perfect cube.

Cube Root Property

The cube root property is like a magic trick that simplifies complex expressions in a snap. It's based on the fundamental understanding that if you have a cube of a number, the cube root of that cube takes you right back to the original number. Clear as mud? Let’s break it down.

Imagine you have a number, let's call it 'a'. If you raise 'a' to the power of 3, written as \(a^3\), you now have 'a cubed'. The cube root property states that \(\sqrt[3]{a^3} = a\). So, when you encounter a cube root with an expression to the power of three inside, like \(\sqrt[3]{5^3}\), you can use this property to state with confidence that the cube root is simply 5.

This property is really useful because it helps you quickly evaluate certain radical expressions without the need for tedious calculations or a calculator.

Radical Expressions

Radical expressions involve roots, like square roots, cube roots, and so on. They're an entire family of mathematical expressions that have a number or expression under a radical sign. When numbers are tucked snugly within the radical sign, \(\sqrt[n]{...}\), they can sometimes present a challenge.

But don’t worry, every family has its patterns, and so do radical expressions. For example, cube roots \(\sqrt[3]{...}\) are looking for what number multiplied by itself three times gives the radicand, the number inside the radical. Recognizing that certain numbers are perfect cubes lets you simplify radical expressions without the sweat and tears.

To simplify a cube root, we seek to express the radicand as a perfect cube. Like in our exercise, \(\sqrt[3]{125}\) simplifies to 5 because 125 is a perfect cube of 5. Always be on the lookout for those perfect squares, cubes, and so forth, because they're your ticket to simplifying radical expressions.