Problem 2 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ \sqrt{27}-\sqrt{75} $$

Step-by-Step Solution

Answer

The result of \( \sqrt{27} - \sqrt{75} \) is \( -2\sqrt{3} \).

Step 1: Simplify Each Square Root Separately

To simplify the square roots, we need to express each number under the square root as a product of a square number and another number.For \( \sqrt{27} \):\[ 27 = 3^3 = 3 \times 3^2 \]This means we can rewrite \( \sqrt{27} \) as:\[ \sqrt{27} = \sqrt{3^2 \times 3} = 3\sqrt{3} \]For \( \sqrt{75} \):\[ 75 = 3 \times 25 = 3 \times 5^2 \]Thus, \( \sqrt{75} \) becomes:\[ \sqrt{75} = \sqrt{3 \times 5^2} = 5\sqrt{3} \]

Step 2: Substitute and Simplify the Expression

Now, substitute the simplified forms back into the original expression:\[ \sqrt{27} - \sqrt{75} = 3\sqrt{3} - 5\sqrt{3} \]Since both terms have a common factor \( \sqrt{3} \), you can factor \( \sqrt{3} \) out:\[ (3-5)\sqrt{3} = -2\sqrt{3} \]

Step 3: Summarize the Result

Putting it all together, the simplified form of the expression \( \sqrt{27} - \sqrt{75} \) is:\[ -2\sqrt{3} \]

Key Concepts

Square Numbers

Understanding square numbers is crucial when simplifying square roots. A square number is the result when a number is multiplied by itself. For instance, 1, 4, 9, 16, and 25 are square numbers because they are 1², 2², 3², 4², and 5², respectively.

Using square numbers helps simplify square roots. When you simplify a square root, you look for factors that are square numbers. For example, in the case of \( \sqrt{27} \), identify 9 as a square number, since 9 is 3 squared. This means \( \sqrt{27} \) can be rewritten as \( \sqrt{3^2 \times 3} = 3\sqrt{3} \).

Recognizing square numbers helps to break down more complex square roots into simpler forms, making calculations easier. Always look out for these numbers as a starting point in your simplification process.

Real Numbers

Real numbers are all the numbers you can think of on the number line. They include positive and negative numbers, whole numbers, fractions, and irrational numbers like square roots. In the context of simplifying square roots, real numbers mean every value you work with is actually possible to represent.

When dealing with square roots, it’s often specified that variables represent positive real numbers. This condition ensures that calculations with roots are straightforward and valid. For example, you can assume \( \sqrt{27} \) and \( \sqrt{75} \) are positive real numbers, without any negatives complicating the situation.

Always remember, working within the realm of real numbers allows you to operate through both rational and irrational numbers in your mathematical exercises, keeping every expression meaningful and correct.

Factoring Square Roots

Factoring square roots is a method that simplifies roots by expressing the number inside the root as a product of simpler numbers. Here's how it works:

• Identify any square number factors inside the square root.
• Separate these square factors from the non-square factors.
• Simplify by taking the square root of the square factors.

Using \( \sqrt{75} \) as an example, notice that 25 is a square factor (since 25 is 5 squared). Thus, \( \sqrt{75} \) can be expressed as \( \sqrt{3 \times 5^2} = 5\sqrt{3} \).

Factoring square roots involves recognizing the square number components. Once factored and simplified, expressions are easier to manage. This method is particularly valuable for solving expressions where you need to add or subtract square roots, as it allows for terms to be combined efficiently, leading to a clean and simplified result.