Problem 23 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt{4 x^{4}} $$

Step-by-Step Solution

Answer

The simplified expression is \(2x^2\).

Step 1: Identify the Square Root Components

The expression given is \( \sqrt{4x^4} \). This expression can be separated into the product of two square roots: \( \sqrt{4} \times \sqrt{x^4} \). This step helps in simplifying each part separately.

Step 2: Simplify the Constant Square Root

We first simplify \( \sqrt{4} \). Since 4 is a perfect square, its square root is 2. Therefore, \( \sqrt{4} = 2 \).

Step 3: Simplify the Variable Square Root

Now, simplify \( \sqrt{x^4} \). Since \( x^4 \) is a perfect square (because \((x^2)^2 = x^4)\), its square root is \( x^2 \). Therefore, \( \sqrt{x^4} = x^2 \).

Step 4: Combine the Simplified Components

Now combine the simplified components: \( 2 \times x^2 \). The simplified expression is \( 2x^2 \).

Key Concepts

Square Roots

Square roots represent a fundamental concept in mathematics. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16. Square roots are often symbolized by the radical sign \( \sqrt{} \).

• The expression inside the square root is called the radicand. In \( \sqrt{4x^4} \), 4\( x^4 \) is the radicand.
• Simplifying square roots involves finding the square root of each factor separately, as shown in \( \sqrt{4} \) and \( \sqrt{x^4} \).
• Square roots of perfect squares (like 4, 9, 16, 25) can be simplified to whole numbers.

Understanding how to separate and simplify each component makes handling larger expressions much easier by breaking them down into manageable parts.

Perfect Squares

Perfect squares are numbers that are the square of an integer. For instance, numbers like 1, 4, 9, 16, and 25 form the sequence of perfect squares, as they equal \( 1^2, 2^2, 3^2, 4^2, \) and \( 5^2 \) respectively. Recognizing these helps in simplifying square roots.

• A perfect square results from multiplying an integer by itself, such as \( 4 = 2 \times 2 \).
• In expressions such as \( \sqrt{x^4} \), since \( x^4 = (x^2)^2 \), it is a perfect square, making it easier to simplify.

This concept is particularly helpful because recognizing perfect squares allows quicker simplification of radical expressions, reducing them to simpler form without relying heavily on calculators.

Variable Exponents

Variable exponents involve expressions where a variable is raised to a power. In our expression, \( x^4 \), 4 is the exponent applied to the variable \( x \). Simplifying variable exponents, especially under a square root, involves thinking about how powers work with multiplication and division.

• When you encounter a variable squared, such as \( (x^2)^2 = x^4 \), this can be re-written under a square root as \( \sqrt{x^4} = x^2 \).
• This shows when doubling the exponent's number inside an even root (like a square root), simplify by halving the exponent.

Understanding how to handle variable exponents in square roots simplifies expressions efficiently and is a vital skill in algebraic manipulation.