Problem 34 - College Algebra

Question

In Exercises \(27-38,\) add or subtract terms whenever possible. $$\sqrt{63 x}-\sqrt{28 x}$$

Step-by-Step Solution

Answer

The simplified form of the original expression \( \sqrt{63x} - \sqrt{28x} \) is \( \sqrt{7x} \)

Step 1: Factor the Radicand

One should first factor the radicand in both terms so as to identify any perfect square factors. \( \sqrt{63x} = \sqrt{9 * 7x} \). Similarly, \( \sqrt{28x} = \sqrt{4 * 7x} \)

Step 2: Simplify the Square Roots

Next is to simplify the square root of the factors. The square root of any perfect square yields a whole number. Hence, \( \sqrt{9 * 7x} = 3 \sqrt{7x}\) and \( \sqrt{4 * 7x} = 2 \sqrt{7x}\)

Step 3: Subtract the Simplified Terms

With the similar terms being identified, the simplified square root expressions can now be subtracted. Hence, \( 3 \sqrt{7x} - 2 \sqrt{7x} = 1 \sqrt{7x} = \sqrt{7x}\)

Key Concepts

Factoring Radicands

Factoring the radicand is a crucial step in simplifying radical expressions. The radicand is the number or expression inside a square root (or other radical). To simplify, we start by breaking down the radicand into its prime factors. It helps to identify any perfect square factors. By factoring, we can often simplify the square root into more manageable terms.For example, with \( \sqrt{63x} \), the number 63 can be factored into \( 9 \times 7 \). Similarly, \( \sqrt{28x} \) can be broken down into \( 4 \times 7 \). By identifying these factors, particularly focusing on perfect squares, we can move to the next step of simplification.

Perfect Square Factors

Perfect square factors are special because their square roots are whole numbers. This characteristic makes them immensely useful when simplifying radical expressions. A perfect square is a number like 4, 9, 16, etc., whose square root results in a whole number.Once we factor the radicand, we look for any perfect squares. For \( 63x \), the perfect square is 9, because \( \sqrt{9} = 3 \). In \( 28x \), the perfect square is 4, since \( \sqrt{4} = 2 \). By taking the square root of these perfect squares, we can reduce the complexity of the expression, leading to simpler terms like \( 3\sqrt{7x} \) and \( 2\sqrt{7x} \).

Subtracting Radicals

Subtracting radicals requires that the terms have identical radicands. Once simplified to this form, we can combine or subtract them just like regular algebraic terms. Think of it as handling like terms in polynomial expressions.In our example, after simplifying, both terms become \( 3\sqrt{7x} \) and \( 2\sqrt{7x} \), which have the same radicand of \( 7x \). This similarity allows us to subtract the coefficients: \( 3 - 2 = 1 \), giving us the final simplified expression \( \sqrt{7x} \).

• Ensure terms have the same radicand before subtracting.
• Simplify each term by extracting whole numbers from any perfect square factors.

This method allows us to efficiently manage and simplify radical expressions through subtraction.