Problem 77 - Intermediate Algebra

Question

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[4]{16 x^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(2x^{1/2}\).

Step 1: Understand the Radical Expression

We have the expression \( \sqrt[4]{16 x^2} \). This is a fourth root operation involving the entire expression \( 16x^2 \). Our goal is to convert this radical expression into an expression with a rational exponent.

Step 2: Express the Radical as an Exponent

The general rule for converting a radical to a rational exponent is \( \sqrt[n]{a} = a^{1/n} \). Applying this rule, the expression \( \sqrt[4]{16 x^2} \) can be written as \( (16 x^2)^{1/4} \).

Step 3: Distribute the Exponent

Using the property \((a \, b)^{m} = a^{m} \, b^{m}\), we distribute the \(1/4\) exponent to each part inside the parenthesis: \((16)^{1/4} \cdot (x^2)^{1/4}\).

Step 4: Simplify Each Component

- First, simplify \(16^{1/4}\). Since \(16\) is \(2^4\), we have \((2^4)^{1/4} = 2^{4 \times 1/4} = 2^1 = 2\).- Next, simplify \((x^2)^{1/4}\) which is \(x^{2 \times 1/4} = x^{1/2}\).

Step 5: Combine Simplified Parts

Combine the simplified parts: We have \(2\) from \(16^{1/4}\) and \(x^{1/2}\) from \((x^2)^{1/4}\). Therefore, the simplified expression is \(2x^{1/2}\).

Key Concepts

Radical Expressions

A radical expression involves roots, such as square roots, cube roots, and, as in our example, fourth roots. Radicals are used when we want to determine a number which, when raised to a certain power, gives us another specified number. In simple terms, it's about finding what number "created" the given result when multiplied a specific number of times.

In the expression \( \sqrt[4]{16x^2} \), the number 4 is known as the index, which tells us that we're dealing with a fourth root. The expression inside the radical sign—here, \( 16x^2 \)—is called the radicand. Our task is to convert this radical expression into another form that is easier to work with: a rational exponent format.

Simplifying Radicals

Simplifying radicals often means converting them into simpler forms, or even as integer numbers, when possible. One popular method is to turn the radicals into rational exponents, which makes multiplication and division much easier.

To simplify \( \sqrt[4]{16x^2} \), we start by expressing the radical as a rational exponent: \( (16x^2)^{1/4} \). Once in this form, distribute the exponent of \( \frac{1}{4} \) to each element inside \( (16x^2) \) separately. It's as if you are handing out pizza slices to a group: each component gets its share. Thus, it becomes \( 16^{1/4} \cdot (x^2)^{1/4} \).

We simplify each term individually: \( 16^{1/4} \) becomes \( 2 \), because \( 16 \) is \( 2^4 \) and the fourth root operation results in \( 2^1 = 2 \). For \( (x^2)^{1/4} \), apply the rule of multiplying exponents for the \( x \): \( x^{2 \times \frac{1}{4}} = x^{1/2} \), giving the simpler form \( x^{1/2} \).

Rational Exponent Rules

Rational exponents are a powerful tool for simplifying radical expressions. They turn the root operation into a fraction, where the numerator is the power and the denominator is the root. For instance, \( a^{m/n} \) means the \( n \)-th root of \( a \) raised to the power of \( m \).

In practice, rules such as \( a^{m} \cdot a^{n} = a^{m+n} \) and \( (a^{m})^n = a^{m \cdot n} \) help manipulate these expressions easily. Particularly, it’s important to know that distribution law for exponents, which is \( (ab)^m = a^m \cdot b^m \), permits the separation of terms under a single power. This principle was shown when we split the expression \( (16x^2)^{1/4} \) into \( 16^{1/4} \cdot (x^2)^{1/4} \).

By using rational exponents, you can simplify complex radicals step by step, achieving cleaner and more understandable expressions.