Problem 35 - College Algebra

Question

In Exercises \(27-38,\) add or subtract terms whenever possible. $$3 \sqrt{18}+5 \sqrt{50}$$

Step-by-Step Solution

Answer

The simplified expression is \(34\sqrt{2}\)

Step 1: Simplifying Square Roots

First, break down the square roots into their simplest form. For \( \sqrt{18} \), we can express it as \( \sqrt{9*2} \) which simplifies as \( 3\sqrt{2} \). Similarly, for \( \sqrt{50} \), we can express it as \( \sqrt{25*2} \) which simplifies to \( 5\sqrt{2} \). So, the original expression becomes \( 3*3\sqrt{2} + 5*5\sqrt{2} = 9\sqrt{2} + 25\sqrt{2} \)

Step 2: Addition of Similar Terms

When you have square roots like \( \sqrt{2} \) in both terms of addition, they can be considered as similar terms. Hence, they can be added together. Perform the operation \( 9\sqrt{2} + 25\sqrt{2} = 34\sqrt{2} \)

Key Concepts

Simplifying Square Roots

When faced with square roots in mathematics, simplifying them can often make calculations easier and more manageable. To simplify a square root, look for factors of the number inside the radical that are perfect squares—numbers like 4, 9, 16, 25, and so on, that have square roots which are whole numbers. For example, with \( \sqrt{18} \), we identify 9 as a perfect square that is a factor of 18. We can then write \( \sqrt{18} \) as \( \sqrt{9 \times 2} \), which simplifies to \( 3\sqrt{2} \) since \( \sqrt{9} = 3 \).

Similarly, \( \sqrt{50} \) can be rewritten as \( \sqrt{25 \times 2} \), which simplifies to \( 5\sqrt{2} \) because \( \sqrt{25} = 5 \). Understanding that simplifying square roots is essentially breaking them down into a product of the square root of a perfect square and the square root of a non-perfect square can significantly ease your way through handling radical expressions in arithmetic.

Radical Expressions

A radical expression involves roots, such as square roots, cube roots, and such. In the case of square roots, the radical sign \( \sqrt{\phantom{a}} \) indicates the operation to find a number, which, when multiplied by itself, gives the number under the root. A radical expression like \( \sqrt{50} \) is expressing the number whose square is 50.

When simplifying radical expressions, it is crucial to find the largest square factor of the number beneath the radical to reduce it to its simplest form, as illustrated in the given exercise. The goal is to express the radical expression as a product of a whole number and a simpler radical, making subsequent operations with these roots, like addition or multiplication, far simpler. Recognizing and handling such expressions is a foundational skill in algebra and one that calls for practice to achieve fluency.

Arithmetic with Square Roots

Performing arithmetic operations with square roots, like addition in our example, requires an understanding of like terms. Similar to how you would combine \(3x\) and \(5x\) into \(8x\), you can combine square roots when the radical part—the number under the square root—is the same. After simplifying \( \sqrt{18} \) and \( \sqrt{50} \) to \( 3\sqrt{2} \) and \( 5\sqrt{2} \), we get terms that can be added together because they share the same \( \sqrt{2} \) part.

Thus, these terms are 'like terms' and can be added to get \( 9\sqrt{2} + 25\sqrt{2} = 34\sqrt{2} \). It's essential to recognize that only the coefficients of the square roots are added while the radical part remains intact. This process relies on the foundational arithmetic property that only like terms can be combined through addition or subtraction, which also leads to a significant simplification of expressions containing square roots.