Problem 24 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(12 \sqrt{3}-4 \sqrt{75}\)

Step-by-Step Solution

Answer

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

Step 1: Identify and Simplify Square Roots

First, we need to simplify each square root in the expression. The term \(\sqrt{75}\) can be simplified as: \(\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}\). Therefore, the expression now becomes: \(12\sqrt{3} - 4(5\sqrt{3})\).

Step 2: Apply the Distributive Property

Next, apply the distributive property to the term \(-4(5\sqrt{3})\) to get \(-20\sqrt{3}\). Now, the expression is \(12\sqrt{3} - 20\sqrt{3}\).

Step 3: Combine Like Terms

Since both terms in the expression have the common part \(\sqrt{3}\), we can combine them. By subtracting the coefficients, we have: \(12 - 20 = -8\). Thus, the expression simplifies to: \(-8\sqrt{3}\).

Key Concepts

Square Roots

Square roots are used to find a number which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9. In simplifying expressions, it's common to encounter square roots like \( \sqrt{75} \). To simplify, we can break down \( \sqrt{75} \) into its prime factors:

• First, note that 75 can be factored into 25 and 3.
• Since 25 is a perfect square, \( \sqrt{25} = 5 \), hence \( \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3} \).

This is a crucial step that helps in making expressions easier to work with. By simplifying the square root, calculations become more straightforward.

Distributive Property

The distributive property is a fundamental principle that allows us to multiply a single term by each term in a parenthesis, effectively "distributing" the multiplication across terms. In the expression \(-4(5\sqrt{3})\), the distributive property helps us break it down into easier parts. Here's how it works:

• We distribute the \(-4\) across \(5\sqrt{3}\).
• This gives us \(-4 \times 5 \times \sqrt{3} = -20\sqrt{3}\).

This method is especially useful when dealing with complex algebraic expressions, helping simplify them efficiently. Using the distributive property, we can take any shared factor outside and make operations cleaner.

Combining Like Terms

Combining like terms is a method used to simplify expressions, bringing together terms that have identical variable parts. To do this, you can simply add or subtract their coefficients. In the expression we have, \(12\sqrt{3} - 20\sqrt{3}\), both terms contain the \(\sqrt{3}\) component. Here's how we proceed:

• Identify the like terms which, in this case, are terms both including \(\sqrt{3}\).
• Since the terms are similar, subtract their coefficients: \(12 - 20 = -8\).

Thus, the expression simplifies to \(-8\sqrt{3}\). Combining like terms not only simplifies expressions but also makes them easier to understand and use in further calculations.