Problem 67 - Algebra A Combined Function

Question

Use rational exponents to simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[4]{(x+3)^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \sqrt{x+3} \).

Step 1: Understand Rational Exponents

Rational exponents allow us to express roots as exponents. The expression \( \sqrt[n]{a^m} \) can be rewritten using rational exponents as \( a^{\frac{m}{n}} \). Here, \( n \) is the root and \( m \) is the power.

Step 2: Identify the Components

In the expression \( \sqrt[4]{(x+3)^{2}} \), the component \( (x+3)^{2} \) is raised to the fourth root. This means \( n = 4 \) and \( m = 2 \).

Step 3: Apply the Rational Exponent Rule

Rewrite the fourth root using rational exponents: \[ \sqrt[4]{(x+3)^{2}} = (x+3)^{\frac{2}{4}} \].

Step 4: Simplify the Fraction Exponent

Simplify the fraction \( \frac{2}{4} \) to its simplest form. This fraction simplifies to \( \frac{1}{2} \).

Step 5: Write the Simplified Expression

The expression with the simplified rational exponent is \( (x+3)^{\frac{1}{2}} \). This means the original expression simplifies to \( \sqrt{x+3} \).

Key Concepts

Radical Simplification

Radical simplification is the process of making a mathematical expression involving roots simpler or easier to manage. It often involves converting radical expressions into forms that are easier to read and work with. For example:

• Finding the root of a number is essentially finding what number multiplied by itself a certain number of times equals the original number.
• Radicals can be simplified by extracting factors that are perfect squares, cubes, etc., which allows you to take them outside the radical sign.

When you encounter a radical expression like \( \sqrt[4]{(x+3)^2} \), simplification can make it easier to handle by converting it to a different form using rational exponents.

Fractional Exponents

Fractional exponents are a handy way to write roots in a more concise manner. They represent both roots and powers in a single form, like \( a^{m/n} \). Here's what you need to know:

• The numerator \( m \) represents the power to which the base is raised.
• The denominator \( n \) represents the root, or the degree of the radical.

For example, \( \sqrt[4]{(x+3)^2} \) can be expressed using fractional exponents as \( (x+3)^{2/4} \). Just by simplifying the fraction, you turn it into a more manageable form: \( (x+3)^{1/2} \). Therefore, fractional exponents simplify dealing with roots and powers by making it straightforward to reduce complex expressions.

Mathematical Expressions

Mathematical expressions are combinations of numbers and variables, connected by operations such as addition, multiplication, and exponentiation. Simplifying them involves:

• Reducing the expression to a more compact or simpler form.
• Making sure expressions are easier to evaluate, especially when substituting specific values.

In the case of the expression \( \sqrt[4]{(x+3)^2} \), simplifying it into \( \sqrt{x+3} \) involves changing the expression using exponent rules. This streamlined expression is not only easier to read but is also more efficient to work with in advanced mathematical problems. Through simplification, you're aiming to convert complex expressions into forms that reveal the underlying relationships and principles.