Problem 22 - Intermediate Algebra

Question

Add or subtract. $$ -\sqrt{75}+\sqrt{12}-3 \sqrt{3} $$

Step-by-Step Solution

Answer

The result is \(-6\sqrt{3}\).

Step 1: Simplify each square root

We start by simplifying the square roots in the expression. \(-\sqrt{75}\) can be simplified as \(-\sqrt{25 \times 3} = -\sqrt{25} \times \sqrt{3} = -5 \sqrt{3}\). \(\sqrt{12}\) simplifies as \(\sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2 \sqrt{3}\).

Step 2: Substitute simplified expressions

Replace the original terms with the simplified square roots in the expression: \(-5 \sqrt{3} + 2 \sqrt{3} - 3 \sqrt{3}\).

Step 3: Combine like terms

Now add and subtract the like terms involving \(\sqrt{3}\). This becomes \((-5 + 2 - 3) \sqrt{3}\).

Step 4: Simplify the coefficients

Calculate the arithmetic: \(-5 + 2 - 3 = -6\). So the expression simplifies to \(-6 \sqrt{3}\).

Key Concepts

Simplifying Square Roots

When working with square roots, the first step is often to simplify them. This involves expressing larger numbers under the square root sign as a product of smaller numbers, preferably including perfect squares. Looking at the example provided, start with

• \(-\sqrt{75}\): Recognize 75 can be broken down into \(25\times3\), where 25 is a perfect square (as \(5^2\)). Then, \(-\sqrt{75} = -\sqrt{25 \times 3} = -\sqrt{25} \times \sqrt{3} = -5\sqrt{3}\).
• \(\sqrt{12}\): Similarly, \(12\) can be expressed as \(4\times3\), with 4 being the perfect square (as \(2^2\)). Thus, simplify \(\sqrt{12}\) to \(\sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}\).

Simplifying square roots like this reduces the term to its simplest form, allowing for easier operations such as addition or subtraction.

Combining Like Terms

After simplifying, it's crucial to identify and combine like terms. Like terms share the same variable part, which in our case is the \(\sqrt{3}\). Once simplified, the expression \(-5\sqrt{3} + 2\sqrt{3} - 3\sqrt{3}\) reveals several like terms.

• The strategy involves adding or subtracting the coefficients (numbers in front of \(\sqrt{3}\)), which are \(-5\), \(+2\), and \(-3\).
• By merging these, focus solely on the coefficients and process as simple arithmetic: \(-5 + 2 - 3\).

Combining these gives a single term with a coefficient adjusted to reflect all contributions: \(-6\sqrt{3}\). This simplification makes the expression more concise and is essential for any calculations involving radicals.

Arithmetic with Radicals

Performing arithmetic operations with radicals follows similar principles to regular arithmetic, but with a focus on preserving the radical's integrity. When dealing with the arithmetic in the expression \(-5\sqrt{3} + 2\sqrt{3} - 3\sqrt{3}\), notice that each term shares the \(\sqrt{3}\).

• This commonality allows us to focus on just the coefficients for addition and subtraction: calculate \(-5 + 2 - 3\).
• Solving: \(-5 + 2 = -3\), and \(-3 - 3 = -6\).

Thus, the expression simplifies to \(-6\sqrt{3}\). When working with radicals, always check if terms can be simplified first, then look for like terms to combine, making sure to perform accurate arithmetic on coefficients. This streamlines the process and ensures clarity in results.