Problem 24 - College Algebra

Question

Simplify each expression. $$12 \sqrt{3}-4 \sqrt{75}$$

Step-by-Step Solution

Answer

The simplified expression is \( -8 \sqrt{3} \).

Step 1: Identify each term

The expression given is \( 12 \sqrt{3} - 4 \sqrt{75} \). We need to simplify each term separately before combining them.

Step 2: Simplify \( \sqrt{75} \)

To simplify \( \sqrt{75} \), we need to factor 75 into its prime factors: \( 75 = 3 \times 5 \times 5 \). So, \( \sqrt{75} = \sqrt{3 \times 5^2} = \sqrt{3} \times 5 = 5 \sqrt{3} \).

Step 3: Simplify \( 4 \sqrt{75} \)

Now, substitute \( \sqrt{75} \) with its simplified form: \( 4 \sqrt{75} = 4 \times 5 \sqrt{3} = 20 \sqrt{3} \).

Step 4: Combine like terms

Rewrite the expression with the simplified terms: \( 12 \sqrt{3} - 20 \sqrt{3} \). Combine the like terms by subtracting the coefficients of \( \sqrt{3} \): \( (12 - 20) \sqrt{3} = -8 \sqrt{3} \).

Step 5: Final result

The simplified expression is \( -8 \sqrt{3} \). This is the simplest form of the original expression.

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This means getting rid of any unnecessary part by clarifying the expression. In the context of radical expressions, simplifying an expression typically includes combining like terms, and breaking down radicals into simpler components.
It starts with identifying each part of the expression. For example, the expression \(12 \sqrt{3} - 4 \sqrt{75}\) consists of two terms: \(12 \sqrt{3}\) and \(4 \sqrt{75}\).
The main goal is to simplify the radicals, if possible, and combine any like terms. This makes the expression easier to work with or interpret.

• Normalize radicals by finding prime factors.
• Factor out common elements from under the radical sign.
• Reevaluate and combine similar items in the expression to their simplest form.

Breaking down these parts into their simplest form is essential to master this concept fully.

Prime Factorization

Prime factorization is a powerful tool used in simplifying radicals. It involves breaking down a number into its prime number factors. These are numbers that are only divisible by themselves and one.
For example, when we simplify \(\sqrt{75}\), we first determine the prime factors of 75. This results in \(75 = 3 \times 5 \times 5\).
By expressing 75 in terms of its prime factors, \(\sqrt{75}\) can be simplified as \(\sqrt{3 \times 5^2} = 5 \sqrt{3}\). This step transforms a complex expression into its simplified form.

• Prime factors help to simplify the radical.
• The square of a prime number factor moves outside the square root symbol.

Utilizing prime factorization allows only non-prime factors to remain under the radical, leading to an easier form of the original expression.

Combining Like Terms

Combining like terms involves simplifying expressions by adding or subtracting terms with the same variables and powers. In radical expressions, this often relates to summing or subtracting coefficients of identical square roots.
In our example, both terms in \(12 \sqrt{3} - 20 \sqrt{3}\) include \(\sqrt{3}\). This makes them 'like terms'. We can therefore manipulate the coefficients. By combining them, we simplify the overall expression.
So, subtracting these coefficients gives us \((12 - 20) \sqrt{3} = -8 \sqrt{3}\).

• Look for common radicals in terms.
• Focus on the coefficients of the identical radicals to combine terms.

This concept is crucial for simplifying expressions efficiently, as it reduces complexity through unified terms.