Problem 40 - Algebra 2 and Trigonometry

Question

In \(38-57,\) write each radical expression as a power with positive exponents and express the answer in simplest form. The variables are positive numbers. $$ \sqrt[3]{12} $$

Step-by-Step Solution

Answer

The expression \( \sqrt[3]{12} \) in exponential form is \( 12^{1/3} \).

Step 1: Identify the Radical Form

The given expression is \( \sqrt[3]{12} \), which represents the cube root of 12. The cube root of any number \( x \) can be written in exponential form as \( x^{1/3} \).

Step 2: Convert the Radical to Exponential Form

Using the relationship \( \sqrt[n]{x} = x^{1/n} \), transform the cube root \( \sqrt[3]{12} \) to its exponential form: \( 12^{1/3} \). This is the expression with a positive exponent.

Step 3: Simplify the Expression Further (if possible)

Check if 12 can be broken down into its prime factors to simplify it further. The prime factorization of 12 is \( 2^2 \times 3^1 \). However, since \( 12^{1/3} \) does not simplify to a simpler form than what is already written, the expression remains \( 12^{1/3} \).

Key Concepts

Cube Roots

Cube roots can be a bit tricky whenever we first encounter them, but they are simply the number that, when multiplied by itself twice more, gives us the original number. In mathematical terms, for any number \( x \), the cube root is the expression \( \sqrt[3]{x} \). This expression asks: "What number, when cubed, results in \( x \)?"

For example, the cube root of 8 is 2 since \( 2 \times 2 \times 2 = 8 \). Similarly, finding the cube root of 27 results in 3, because \( 3 \times 3 \times 3 = 27 \).

Understanding the cube root allows us to easily convert it into an exponential form. If you need the cube root of a number like 12, you can express it as \( 12^{1/3} \). This representation often helps simplify expressions, particularly during more complex mathematical equations.

Exponents

Exponents represent repeated multiplication of a number by itself. When you see an expression like \( x^n \), it tells you to multiply \( x \) by itself \( n \) times. For instance, \( 3^2 \) is simply \( 3 \times 3 = 9 \).

Exponents are also helpful when dealing with root values. The link between exponents and roots is the fractional exponent.

• The square root is represented as an exponent of \( \frac{1}{2} \) — so \( \sqrt{x} = x^{1/2} \).
• The cube root, as seen with \( \sqrt[3]{12} \), can be expressed as \( 12^{1/3} \).

This conversion to an exponential form is especially useful for simplifying and manipulating expressions involving roots. It offers a straightforward way to manage calculations without dealing directly with roots, which can be more challenging.

When simplified correctly, exponential notation brings clarity and simplicity, turning complex root expressions into manageable exponent forms.

Simplifying Expressions

Simplifying expressions is a key technique in mathematics for making these expressions easier to understand and work with. The goal of simplification is to reduce an expression to its most basic form.

When dealing with radical expressions, a useful step is to convert them into exponents, as we showed for the cube root: \( \sqrt[3]{12} \) becomes \( 12^{1/3} \). This form allows you to explore further possibilities for simplification.

To simplify an expression like \( 12^{1/3} \), you can use prime factorization:

• Break down 12 into its prime factors: \( 12 = 2^2 \times 3 \).
• Apply the exponent to each prime factor separately: \( (2^2)^{1/3} \times (3^1)^{1/3} \).

Despite using prime factorization, sometimes an expression cannot be further simplified beyond its current form.

Always aim to express the number in the simplest form, and remember, a key part of mathematics is understanding when something is as reduced as it can be. In our example, \( 12^{1/3} \) remains the simplest form.