Problem 21 - A Graphical Approach to College Algebra

Question

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt[3]{125}$$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt[3]{125}\) is 5.

Step 1: Identify the Cube Root Expression

The expression given is \(\sqrt[3]{125}\), which signifies the cube root of 125. Our task is to simplify this cube root.

Step 2: Factor the Radicand into Prime Factors

125 is the number under the cube root sign. We need to factor 125 into its prime factors. 125 can be decomposed as \(5 \times 5 \times 5 \), or, more compactly, \(5^3\).

Step 3: Apply the Cube Root to the Prime Factorization

Since 125 is \(5^3\), the cube root of \(5^3\) is \(5\). The cube root and the exponent cancel each other out as the exponent is exactly 3, giving us: \(\sqrt[3]{5^3} = 5\).

Step 4: Simplify the Expression

The simplification gives us the final result, which is just the base of the original exponentiation, \(5\). This is the simplified form of \(\sqrt[3]{125}\).

Key Concepts

Cube Roots

When we talk about cube roots, we're referring to the process of finding a number that, when multiplied by itself three times, gives us the original number. For example, the cube root of 8 is 2 because when you multiply 2 by itself three times (2 × 2 × 2), you get 8.
This concept is a fundamental part of working with radical expressions and helps us simplify numbers that are raised to the power of 3.

• Cube roots are represented by the symbol \(\sqrt[3]{...}\).
• The expression inside the cube root symbol is called the radicand.

In our exercise, the radicand is 125, and our task is to simplify \(\sqrt[3]{125}\). Finding cube roots often involves prime factorization.

Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime factors. Prime numbers are those greater than 1 that have no divisors other than 1 and themselves, like 2, 3, 5, 7, etc.
When we factor a number, we break it down until we can't divide anymore except by 1 or the number itself.

• For instance, to find the prime factors of 18, we'd divide it into \(2 \times 3 \times 3\), or equivalently \(2 \times 3^2\).
• This method is really helpful when simplifying radical expressions.

In our example of \(\sqrt[3]{125}\), we first determine the prime factors of 125. By dividing, we find that 125 is \(5 \times 5 \times 5\). That can be written as \(5^3\). This prime factorization makes applying the cube root straightforward.

Exponents

Exponents are a way to express a number multiplied by itself a certain number of times. They are written as small raised numbers to the right of the base number.
For example, \(3^2\) means 3 multiplied by itself, which equals 9.

• The base is the number being multiplied, and the exponent tells you how many times to multiply the base by itself.
• Exponents are extremely useful for simplifying expressions and working with large numbers.

In the context of cube roots, like \(\sqrt[3]{5^3}\), exponents tell us how many times the number is used in multiplication. When the exponent matches the root, such as in \(\sqrt[3]{5^3}\), it simplifies the operation, giving us just the base number, which is 5. This simplicity is one of the beauties of working with exponents.