Problem 52 - College Algebra

Question

Simplify each expression. $$\sqrt{108 x^{4}}+\sqrt{27 x^{4}}$$

Step-by-Step Solution

Answer

The expression simplifies to \(9x^2 \sqrt{3}\).

Step 1: Simplify the First Square Root

First, simplify the square root of \(108x^4\). Start by factoring \(108\) as \(2^2 \times 3^3\). So, we have \(\sqrt{2^2 \times 3^3 \times x^4}\). Since \(x^4\) is a perfect square, it comes outside the square root as \(x^2\), and \(\sqrt{2^2}\) comes outside as \(2\). Inside the square root, \(3^3\) simplifies to \(3 \sqrt{3}\). Thus, \(\sqrt{108x^4} = 6x^2 \sqrt{3}\).

Step 2: Simplify the Second Square Root

Now simplify \(\sqrt{27x^4}\). Factor \(27\) as \(3^3\). We have \(\sqrt{3^3 \times x^4}\). Again, \(x^4\) is a perfect square and comes outside as \(x^2\), and \(\sqrt{3^2} = 3\). This leaves us with \(\sqrt{3}\) inside. So, \(\sqrt{27x^4} = 3x^2 \sqrt{3}\).

Step 3: Combine the Expressions

After simplifying both square roots, we have: \(6x^2 \sqrt{3} + 3x^2 \sqrt{3}\). Factor out the common term, \(x^2 \sqrt{3}\), from both terms: \((6 + 3)x^2 \sqrt{3} = 9x^2 \sqrt{3}\).

Step 4: Final Expression

The expression simplifies to \(9x^2 \sqrt{3}\). This is the simplified form of the original expression.

Key Concepts

Factorization

When dealing with algebraic expressions, factorization is a powerful tool. It involves breaking down an expression into a product of simpler expressions, or factors, that when multiplied together give the original expression. The goal is to represent expressions in a simpler form. It's particularly useful when simplifying expressions that include square roots and other operations.

In the exercise given, we first tackled the factorization of numbers. For example, the number 108 was broken down into its prime factors: \(2^2 \times 3^3\). Factorization can also apply to variables, especially when they are raised to a power, as with \(x^4\). By identifying how numbers and variables can be represented by their factors, it becomes much easier to simplify square roots and other expressions.

Remember, when you're working with factorization, focus on:

• Finding prime factors of the numerical part
• Breaking down powerful terms like \(x^n\) into smaller, manageable components, such as recognizing \(x^4\) as \((x^2)^2\)
• Using these factors to simplify further parts of the expression, such as square roots

Factorization makes the process of simplification much more manageable, especially in complex algebraic scenarios.

Square Roots

Square roots are an essential part of algebra. The square root of a number \(n\) is a value that when multiplied by itself gives \(n\). In algebra, square roots often accompany variable and numerical expressions, requiring specific methods to simplify them.

In our exercise, we encounter expressions like \(\sqrt{108x^4}\). The goal is to simplify it by separating perfect squares, which can be removed from under the square root. For instance, \(x^4\) is already a perfect square (since it is \((x^2)^2\)), allowing it to be taken outside the root as \(x^2\). Similarly, the factor \(2^2\) is a perfect square, allowing it to move outside the root as the number 2, simplifying the expression further.

Key points when simplifying square roots include:

• Identifying and extracting perfect squares from the expression
• Using basic properties of square roots, such as \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)
• Always simplifying completely by combining like terms

This strategy of breaking down square roots into simpler parts allows for easier manipulation and combination of terms across algebraic expressions.

Perfect Squares

A perfect square is any number or expression that can be written as a product of an integer or expression with itself. For example, the number 9 is a perfect square because it is \(3^2\), and similarly, \(x^4\) is a perfect square as it is \((x^2)^2\). Recognizing perfect squares is crucial when simplifying algebraic expressions.

When you identify perfect squares within a square root or other mathematical expression, you can simplify it by taking the square outside the square root. This significantly reduces the complexity of the expression. For example, in the exercise, \(x^4\) inside the square root simplifies the expression by letting \(x^2\) come outside, as it is a perfect square.

To effectively handle perfect squares, consider:

• Spotting numbers and variables that are perfect squares
• Expressing them in a squared form to simplify the calculation
• Removing perfect squares from the square root as integer or simpler variable factors

Understanding how to manipulate perfect squares enables the reduction of complex algebraic expressions to their simplest forms, leading to efficient solutions.