Problem 31 - Algebra 2 and Trigonometry

Question

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{9 x}+\sqrt{25 x} $$

Step-by-Step Solution

Answer

The simplest form is \( 8\sqrt{x} \).

Step 1: Simplify each radical

Identify the perfect squares in each radical expression. For \( \sqrt{9x} \), recognize that \(9\) is a perfect square, so extract \(3\) from the square root: \( \sqrt{9x} = 3\sqrt{x} \). Similarly, for \( \sqrt{25x} \), extract \(5\) because \(25\) is a perfect square: \( \sqrt{25x} = 5\sqrt{x} \).

Step 2: Combine like terms

Now that both expressions are simplified, combine the like radical terms. Both terms share \( \sqrt{x} \), making them like terms. Combine them as such: \( 3\sqrt{x} + 5\sqrt{x} = (3 + 5)\sqrt{x} = 8\sqrt{x} \).

Key Concepts

Perfect Squares

Understanding perfect squares is essential when simplifying radicals. A perfect square is the product of an integer multiplied by itself. For instance, numbers like 1, 4, 9, 16, and 25 are perfect squares because they can be expressed as \(1^2, 2^2, 3^2, 4^2,\) and \(5^2\) respectively.

When dealing with radical expressions, identifying perfect squares within the radicand (the expression inside the square root) allows for easier simplification. When you encounter a perfect square within a square root, you can "extract" the square root of that number. For example, \(\sqrt{9} = 3\) because 9 is a perfect square. Similarly, \(\sqrt{25} = 5\).

By recognizing and extracting these perfect squares, you can significantly simplify the initial expressions, paving the way for further simplification.

Like Terms

The concept of like terms is crucial when combining expressions in algebra. Like terms are terms that have the same variable raised to the same power. This makes them eligible for combining through addition or subtraction.

In the context of our radical expression \(3\sqrt{x} + 5\sqrt{x}\), both terms are like terms because they contain the same radical \(\sqrt{x}\). They differ only in their coefficients (3 and 5, respectively).

To combine like terms, simply add or subtract their coefficients, maintaining the variable part. For our example, \(3\sqrt{x} + 5\sqrt{x}\) simplifies to \((3 + 5)\sqrt{x} = 8\sqrt{x}\).

This process simplifies complex expressions and is a fundamental aspect of algebraic manipulation.

Square Root Simplification

Simplifying square roots involves transforming them into their simplest form. This means reducing the square root to a product of simpler numbers whenever possible.

To start, identify any perfect square factors within the radicand. Factor them out to simplify the square root. Consider our original expressions \(\sqrt{9x}\) and \(\sqrt{25x}\) as examples. Recognize that 9 and 25 are perfect squares. Extract these to get \(3\sqrt{x}\) and \(5\sqrt{x}\), respectively.

After simplifying each square root by removing the perfect square factors, you'll often be left with a smaller, equivalent expression. The next step is to check whether any like terms can be combined, further simplifying the given expression.

• Identify perfect square factors.
• Extract simplified square roots where possible.
• Combine like terms, if applicable.

By following these steps, you can transform complex expressions into a more manageable and understandable form, ensuring clarity in mathematical problem-solving.