Problem 191 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{1}{2} \sqrt{20}-\frac{2}{3} \sqrt{45} $$

Step-by-Step Solution

Answer

-\sqrt{5}

Step 1: Simplify the square roots

Simplify the square roots in the expression. Notice that \( \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} \) and \( \sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5} \). So the expression becomes \( \frac{1}{2} \cdot 2\sqrt{5} - \frac{2}{3} \cdot 3\sqrt{5} \).

Step 2: Simplify the coefficients

Simplify the coefficients in front of the square roots. We get \( \frac{1}{2} \cdot 2\sqrt{5} = \sqrt{5} \) and \( \frac{2}{3} \cdot 3\sqrt{5} = 2\sqrt{5} \).

Step 3: Combine like terms

Combine the like terms to simplify the expression. We have \( \sqrt{5} - 2\sqrt{5} = -\sqrt{5} \).

Key Concepts

square roots

Square roots are a fundamental concept in algebra and help us simplify expressions. A square root asks you what number, when multiplied by itself, will give the original number. For example, the square root of 16 is 4 because 4 times 4 equals 16.
When simplifying expressions involving square roots, look for factors that are perfect squares. These make it easier to simplify. In our exercise, we have: \[ \sqrt{20} \quad \text{and} \quad \sqrt{45} \]
We can break these down into smaller components: \[ \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} \quad \text{and} \quad \sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5} \] Through these steps, the expression becomes simpler and easier to work with.

like terms

In algebra, 'like terms' are terms that have the same variable raised to the same power. They can be combined to make expressions simpler. Identifying and combining like terms is a crucial skill.
For example, in the expression \[ \frac{1}{2} \cdot 2\sqrt{5} - \frac{2}{3} \cdot 3\sqrt{5} \]
Both terms include \( \sqrt{5} \), making them like terms. By combining them, we explore: \[ \sqrt{5} - 2\sqrt{5} = -\sqrt{5} \]
Simplifying expressions by combining like terms allows us to find more straightforward results. Always look for terms that share the same variables for quick identification.

coefficients

Coefficients are numbers in front of variables or square roots in algebraic expressions. They tell us how many times the variable is being multiplied. In the exercise, \[ \sqrt{5} \quad \text{or} \quad 2\sqrt{5} \]
The numbers 1 and 2 are coefficients, respectively. When we simplify our exercise by multiplying and combining like terms, we deal directly with these coefficients.
Here’s the step where coefficients come into play: \[ \frac{1}{2} \cdot 2 = 1 \quad \text{and} \quad \frac{2}{3} \cdot 3 = 2 \]
So the terms simplify to 1\( \sqrt{5} \) and 2\( \sqrt{5} \). Always ensure coefficients are correctly multiplied and simplified before combining like terms.