Problem 56 - Intermediate Algebra

Question

Simplify each radical. Assume that all variables represent positive real numbers. $$ -\sqrt[3]{125} $$

Step-by-Step Solution

Answer

The simplified expression is -5.

Step 1: Identify the radicand

The radical to simplify is \(-\sqrt[3]{125}\). Here, the radicand is 125, which is the number inside the cube root symbol.

Step 2: Express the radicand as a power of a base number

Recognize that 125 can be expressed as a power of 5, specifically \(125 = 5^3\). This means 125 is a perfect cube.

Step 3: Simplify the radical using cube root property

Apply the cube root rule, which states that \(\sqrt[3]{a^3} = a\). Thus, \(\sqrt[3]{5^3} = 5\).

Step 4: Apply the negative sign in front of the radical

Since there is a negative sign before the cube root, apply it to the simplified radical result. Therefore, \(-\sqrt[3]{5^3} = -5\).

Step 5: Write the final simplified expression

The expression simplifies to \(-5\).

Key Concepts

Cube Roots

Cube roots are a fundamental concept in mathematics often used to simplify radicals, just like square roots. A cube root of a number is a special number that, when multiplied by itself three times, equals the original number. For example, with the radical \(\sqrt[3]{125}\), we are seeking a number that, when multiplied by itself three times, will give 125.
This is different from a square root, which involves multiplication twice. Understanding cube roots can help simplify expressions by reducing them to simpler values.

• In the context of the given problem, identifying that 125 is \(5^3\) means recognizing 5 as the cube root of 125.
• Applying the rule \(\sqrt[3]{a^3} = a\) simplifies the radical.

Mastering cube roots is helpful not only in solving algebraic expressions but also when tackling geometry and physics problems involving three-dimensional spaces.

Perfect Cubes

Perfect cubes are numbers that can be expressed as the cube of an integer. This is a key idea in simplifying cube roots because they allow radicals to be simplified more easily. For example, numbers like 8 (\(2^3\)), 27 (\(3^3\)), and 125 (\(5^3\)) are all perfect cubes.
Recognizing perfect cubes can make your calculations smoother, because it means you can bypass finding the cube root, which is particularly useful for quick simplifications.

• A perfect cube is a result of raising an integer to the power of three.
• Remembering common perfect cubes can save time: note numbers like 1 (\(1^3\)), 64 (\(4^3\)), and 216 (\(6^3\)).

When simplifying radicals or working with cube roots, identifying and understanding perfect cubes will serve as a much-needed shortcut.

Negative Sign in Radicals

Handling a negative sign in radicals can sometimes seem confusing, but with cube roots, it's usually more straightforward than with square roots. This is because cube roots can be negative, unlike square roots which must remain non-negative when dealing with real numbers.
In the expression \( -\sqrt[3]{125}\), the negative sign in front of the radical can be carried through the simplification process without any special alterations.

• Cube roots of negative numbers exist: for instance, \(\sqrt[3]{-8} = -2\).
• The negative sign does not affect the basic properties of finding the cube root but does affect the final sign of the result.

When simplifying radicals, keep the negative sign in place unless the problem context indicates otherwise. This understanding ensures your mathematical expressions are both correct and meaningful.