Problem 22 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ -\sqrt{75}+\sqrt{12}-3 \sqrt{3} $$

Step-by-Step Solution

Answer

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

Step 1: Simplify \( -\sqrt{75} \)

First, we simplify \( -\sqrt{75} \). Break 75 into its prime factors: \( 75 = 3 \times 5^2 \). Thus, \( \sqrt{75} = \sqrt{3 \times 5^2} \). By taking 5 out of the square root, we have \( \sqrt{75} = 5\sqrt{3} \). Therefore, \( -\sqrt{75} = -5\sqrt{3} \).

Step 2: Simplify \( \sqrt{12} \)

Now, simplify \( \sqrt{12} \). Break 12 into its prime factors: \( 12 = 2^2 \times 3 \). Thus, \( \sqrt{12} = \sqrt{2^2 \times 3} \). By taking 2 out of the square root, we have \( \sqrt{12} = 2\sqrt{3} \).

Step 3: Substitute Back into Expression

Substitute the simplified parts back into the expression: \( -\sqrt{75} + \sqrt{12} - 3\sqrt{3} \) becomes \( -5\sqrt{3} + 2\sqrt{3} - 3\sqrt{3} \).

Step 4: Combine Like Terms

We now have the expression \( -5\sqrt{3} + 2\sqrt{3} - 3\sqrt{3} \). Combine the coefficients of \( \sqrt{3} \) together: \( -5 + 2 - 3 = -6 \). Thus, the expression becomes \( -6\sqrt{3} \).

Key Concepts

Square Root Calculations

Square root calculations can be puzzling at first, but they are made easier when you understand the process of simplification. The main idea is to identify numbers under the square root (radicands) that can be broken down into their prime factors. Let's take on the example from our exercise, the square root of 75.In this particular problem, we break down the number 75 into its prime factors:

• 75 can be expressed as \( 3 \times 5^2 \).

Using prime factorization, we identify that \( 5^2 \) has a square, so it can "come out" from under the square root.This means we can simplify \( \sqrt{75} \) by taking 5 out, leaving us with \( 5\sqrt{3} \). Thus, \( -\sqrt{75} \) turns into \( -5\sqrt{3} \). Understanding these calculations helps us manage even the complex expressions with more ease. Keeping calculations simple is the key, and this example shows the utility of identifying the square roots of numbers effectively.

Prime Factorization

Prime factorization is a deeply useful tool in mathematics, especially when simplifying radicals. It involves breaking down a number into the smallest possible prime numbers that multiply together to give the original number.Let's examine how this works with our exercise problem:

• For the number 75: The prime factors are \( 3 \times 5^2 \).
• For the number 12: The prime factors are \( 2^2 \times 3 \).

Recognizing these components allows for simplification. For instance, any repeated factors can be taken out of the square root.In \( 12 = 2^2 \times 3 \), we took \( 2^2 \), and extracted \( 2 \), resulting in \( 2\sqrt{3} \). Prime factorization simplifies the expression to a more manageable form by reducing the radicals into understandable terms. Once you grasp the primes, simplifying any square root becomes a routine task.

Combining Like Terms

Combining like terms is an essential step in simplifying expressions with radicals. It is all about recognizing terms that have the same variable or radical part and working on their coefficients.Take the expression after simplification:

• \( -5\sqrt{3} + 2\sqrt{3} - 3\sqrt{3} \)

Each term has \( \sqrt{3} \), indicating they are like terms. To simplify, focus solely on the coefficients—the numbers multiplying \( \sqrt{3} \):

• Start with the existing coefficients: \( -5 + 2 - 3 \).

Add these coefficients together:

• The calculation: \( -5 + 2 - 3 = -6 \).

Thus, the expression simplifies to \( -6\sqrt{3} \). By combining the like terms, expressions are simplified to their most reduced form. Mastery in recognizing and combining like terms prevents errors and leads to the correct solution efficiently.