Problem 189 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{2}{3} \sqrt{72}+\frac{1}{5} \sqrt{50} $$

Step-by-Step Solution

Answer

\( 5 \sqrt{2} \)

Step 1: Simplify the Radicals

First, simplify each square root individually. We know that \ \ \( \sqrt{72} \ = \ \sqrt{36 \times 2} \ = \ 6 \sqrt{2} \) and \ \( \sqrt{50} \ = \ \sqrt{25 \times 2} \ = \ 5 \sqrt{2} \)

Step 2: Substitute Simplified Radicals Back into the Expression

Replace \( \sqrt{72} \) and \( \sqrt{50} \) in the original expression with their simplified forms. \ \ We get: \ \ \[ \frac{2}{3} \sqrt{72} \ + \ \frac{1}{5} \sqrt{50} \ = \frac{2}{3} \cdot 6 \sqrt{2} \ + \ \frac{1}{5} \cdot 5 \sqrt{2} \ = \ 4 \sqrt{2} \ + \ \sqrt{2} \]

Step 3: Combine Like Terms

Since both terms contain \( \sqrt{2} \), they are like terms and can be combined. \ \ Thus, \ \[ 4 \sqrt{2} \ + \ \sqrt{2} \ = \ 5 \sqrt{2} \]

Key Concepts

Square Roots

Square roots are an essential concept in mathematics, particularly in algebra. When you take the square root of a number, you are finding a value that, when multiplied by itself, gives the original number. For example, \(\sqrt{36} = 6\) because \(6 \times 6 = 36\).

In algebraic expressions, it's often necessary to simplify square roots to make calculations easier. For instance, in the problem \(\frac{2}{3} \sqrt{72} + \frac{1}{5} \sqrt{50}\), the square roots of \(72\) and \(50\) are simplified by breaking them down into their prime factors: \(\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}\) and \(\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\).

This simplification makes it easier to combine and further process the terms in an algebraic expression.

Like Terms

Like terms are terms that contain the same variables raised to the same power. In algebra, only like terms can be combined. For example, \(5x\) and \(3x\) are like terms because they both contain the variable \(x\) raised to the first power.

In the exercise \(\frac{2}{3} \sqrt{72} + \frac{1}{5} \sqrt{50}\), after simplifying the square roots to get \(6\sqrt{2}\) and \(5\sqrt{2}\), the resulting expression is \(4\sqrt{2} + \sqrt{2}\).

Since both terms contain \(\sqrt{2}\), they are like terms. Combining like terms involves adding or subtracting the coefficients, which results in \(5\sqrt{2}\).

Algebraic Simplification

Algebraic simplification helps to make an expression easier to work with by combining like terms, simplifying radicals, and using arithmetic operations. This process follows a set of rules to ensure that the expression remains equivalent to its original form.

In the given exercise, simplification involves several steps. First, each square root is simplified: \(\sqrt{72} = 6\sqrt{2}\) and \(\sqrt{50} = 5\sqrt{2}\). Then, these simplified radicals are substituted back into the expression. The expression becomes \(4\sqrt{2} + \sqrt{2}\), which means \(\frac{2}{3} \times 6\sqrt{2} + \frac{1}{5} \times 5\sqrt{2}\).

Next, we combine the like terms. Since both terms include \(\sqrt{2}\), it's straightforward. Add the coefficients together (4 and 1) to get the final simplified expression: \(5\sqrt{2}\).

By simplifying radicals and combining like terms through algebraic simplification, complex expressions can often be reduced to much simpler forms, making them easier to handle in further calculations.