Problem 58 - College Algebra

Question

For the following exercises, simplify each expression. $$ \sqrt{72 c}-2 \sqrt{2 c} $$

Step-by-Step Solution

Answer

The simplified expression is \( 4\sqrt{2c} \).

Step 1: Simplify the First Radicals

First, we need to simplify the square root of 72c. Notice that 72 is the same as 36 times 2, and since 36 is a perfect square, we can simplify this as follows:\[sqrt{72c} = \sqrt{36 \cdot 2c} = \sqrt{36} \cdot \sqrt{2c} = 6\sqrt{2c}.\]

Step 2: Combine Like Terms

Now that we have \( 6\sqrt{2c} \) from the first term and \( -2\sqrt{2c} \) from the second term, combine these like terms:\[6\sqrt{2c} - 2\sqrt{2c} = \left(6 - 2\right)\sqrt{2c} = 4\sqrt{2c}.\]

Key Concepts

Simplifying Radicals

When you simplify radicals, you break them down to their simplest form. It often involves finding perfect squares within the radical. This simplifies the expression and makes calculations more manageable.
For example,

• Consider \( \sqrt{72c} \). Recognize 72 as \( 36 \times 2 \), where 36 is a perfect square.
• Extracting the square root of 36, which is 6, allows us to simplify \( \sqrt{72c} \) to \( 6\sqrt{2c} \).

By identifying and simplifying these radicals, you reduce the complexity of expressions, making them easier to combine or further manipulate in calculations.

Combining Like Terms

Combining like terms is a fundamental operation in algebra that helps in simplifying expressions. It involves merging terms that have the same variable part.
For radicals such as \( 6\sqrt{2c} - 2\sqrt{2c} \), both terms contain the "like" radical expression \( \sqrt{2c} \). This similarity allows you to approach them as if you were handling simple algebraic terms like \( x \):

• Treat \( \sqrt{2c} \) as a single unit.
• Simply subtract the coefficients: \( 6 - 2 = 4 \)

The result of combining these like terms is \( 4\sqrt{2c} \), a simplified version of the original radical expression that is clearer and much easier to work with.

Square Roots

Square roots are fundamental in understanding radicals and expressions that include them. The key idea is finding a number which when multiplied by itself gives the original number under the root.
Understanding square roots involves:

• Identifying perfect squares, like \( 1, 4, 9, 16, 25, \) and so on, where each of these has a whole number as the square root.
• Extracting these square roots simplifies expressions; for example, \( \sqrt{36} = 6 \).

This ability to recognize and utilize square roots enables you to handle larger mathematical problems with greater ease. Simplifying terms like \( \sqrt{72c} \) becomes a breeze when recognizing perfect squares and correctly applying square root rules.