Problem 10 - Algebra for College Students

Question

Use the distributive property to help simplify each of the following. \(13 \sqrt{28}-2 \sqrt{63}-7 \sqrt{7}\)

Step-by-Step Solution

Answer

The expression simplifies to \( 13\sqrt{7} \).

Step 1: Simplify Each Square Root

Start by simplifying each square root in the expression. For \( \sqrt{28} \), note that 28 can be written as \( 4 \times 7 \), so \( \sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7} = 2\sqrt{7} \). Similarly, for \( \sqrt{63} \), we have \( 63 = 9 \times 7 \), hence \( \sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3\sqrt{7} \). The term \( \sqrt{7} \) is already in its simplest form.

Step 2: Substitute Simplified Square Roots

Use the simplified forms found in Step 1 to rewrite the original expression. Substitute \( 2\sqrt{7} \) for \( \sqrt{28} \) and \( 3\sqrt{7} \) for \( \sqrt{63} \). This gives us \( 13(2\sqrt{7}) - 2(3\sqrt{7}) - 7\sqrt{7} \).

Step 3: Apply the Distributive Property

Apply the distributive property to each term. The expression becomes \( 13 \cdot 2 \sqrt{7} - 2 \cdot 3 \sqrt{7} - 7 \cdot \sqrt{7} \), which simplifies to \( 26\sqrt{7} - 6\sqrt{7} - 7\sqrt{7} \).

Step 4: Combine Like Terms

Now, combine the like terms from Step 3. All terms are multiples of \( \sqrt{7} \), so we can sum the coefficients: \( 26 - 6 - 7 = 13 \). Thus, the expression simplifies to \( 13\sqrt{7} \).

Key Concepts

Simplifying Square Roots

When faced with square roots in an algebraic expression, simplifying them can make the problem easier to handle. Let's take a look at how this works through an example:

• The square root of 28: Notice that 28 can be broken down into the product of 4 and 7, which allows us to express the square root of 28 as \( \sqrt{4 \times 7} \).
• Because 4 is a perfect square, we can simplify \( \sqrt{4} \) into 2. Thus, \( \sqrt{28} = 2\sqrt{7} \).
• Similarly, for 63, which can be expressed as \( 9 \times 7 \), we find that \( \sqrt{63} = 3\sqrt{7} \), since 9 is a perfect square and equal to 3 when square rooted.

These steps help break down complex expressions into manageable parts, setting the stage for further simplification.

Like Terms

Like terms are terms in an expression that have the same variables raised to the same power. Identifying these terms is crucial in simplifying expressions. For our purpose:

• In the expression \( 13\sqrt{28} - 2\sqrt{63} - 7\sqrt{7} \), after simplifying the square roots, the expression becomes \( 26\sqrt{7} - 6\sqrt{7} - 7\sqrt{7} \).
• Here, all parts of the expression involve the term \( \sqrt{7} \), which makes them like terms.

Recognizing like terms allows you to combine them by focusing on their numerical coefficients, simplifying the expression further.

Combining Coefficients

One critical step in simplifying algebraic expressions involves combining coefficients. After identifying like terms, the coefficients (numerical parts) must be added or subtracted. Consider the expression after simplification:

• In \( 26\sqrt{7} - 6\sqrt{7} - 7\sqrt{7} \), each term includes the factor \( \sqrt{7} \).
• The coefficients are 26, -6, and -7. Combine these by performing the arithmetic: \( 26 - 6 - 7 = 13 \).

This results in the simplified expression \( 13\sqrt{7} \). By focusing on the coefficients, simplification becomes straightforward, aligning with the simplicity of combining like amounts.

Algebraic Expressions

An algebraic expression encompasses numbers, variables, and operations like addition and subtraction. Simplification involves reducing these expressions to their simplest forms by applying various mathematical properties. The following highlights the process:

• Begin with understanding the composition of the expression. For example: \( 13\sqrt{28} - 2\sqrt{63} - 7\sqrt{7} \).
• By applying properties such as the distributive property, square roots are simplified, and coefficients combined.
• This leads to a final, simpler form: \( 13\sqrt{7} \).

Simplifying algebraic expressions requires recognizing common patterns and simplification opportunities, ultimately aiding problem-solving efficiency.