Problem 11 - Algebra for College Students

Question

Use the distributive property to help simplify each of the following. \(\frac{3}{4} \sqrt{7}-\frac{2}{3} \sqrt{28}\)

Step-by-Step Solution

Answer

The expression simplifies to \(-\frac{7}{12} \sqrt{7}\).

Step 1: Identify Constants and Radicals

We have two terms: \( \frac{3}{4} \sqrt{7} \) and \( -\frac{2}{3} \sqrt{28} \). Notice that both contain radicals (\( \sqrt{7} \) and \( \sqrt{28} \)). We need to simplify where possible by using the distributive property and further operations.

Step 2: Simplify the Radical \( \sqrt{28} \)

The term \( \sqrt{28} \) can be simplified. Since \( 28 = 4 \times 7 \), we can write \( \sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \cdot \sqrt{7} = 2\sqrt{7} \). Thus, \( -\frac{2}{3} \sqrt{28} \) becomes \( -\frac{2}{3} \times 2\sqrt{7} = -\frac{4}{3} \sqrt{7} \).

Step 3: Combine Like Terms

Now we have \( \frac{3}{4} \sqrt{7} - \frac{4}{3} \sqrt{7} \). Since \( \sqrt{7} \) is a common factor, factor it out: \( \left( \frac{3}{4} - \frac{4}{3} \right) \sqrt{7} \).

Step 4: Find the Common Denominator

To simplify \( \frac{3}{4} - \frac{4}{3} \), find the common denominator, which is 12. Convert each fraction: \( \frac{3}{4} = \frac{9}{12} \) and \( \frac{4}{3} = \frac{16}{12} \). The expression becomes \( \left( \frac{9}{12} - \frac{16}{12} \right) \sqrt{7} \).

Step 5: Perform the Subtraction

Subtract the numerators: \( \frac{9}{12} - \frac{16}{12} = \frac{-7}{12} \). So the expression simplifies to \( -\frac{7}{12} \sqrt{7} \).

Key Concepts

Simplifying Radicals

Simplifying radicals involves breaking down a radical expression into its simplest form. In this exercise, we begin by simplifying \( \sqrt{28} \). The number 28 can be factored into 4 and 7, as 28 is the product of these two integers. When we take the square root of this product, we separate it into \( \sqrt{4} \cdot \sqrt{7} \).

• Since \( \sqrt{4} = 2 \), the expression \( \sqrt{28} \) simplifies to \( 2\sqrt{7} \).
• This simplification reduces the complexity of the original term \( -\frac{2}{3} \sqrt{28} \) to \( -\frac{4}{3} \sqrt{7} \).

Simplifying the radical helps in the process of combining like terms later on, as it makes the terms more comparable and easier to work with.

Combining Like Terms

Combining like terms is a crucial step in algebra, which simplifies expressions by adding or subtracting terms with the same variable part. In this exercise, once we've simplified the radical, we end up with two terms: \( \frac{3}{4} \sqrt{7} \) and \( -\frac{4}{3} \sqrt{7} \).

• Both terms share the common radical part \( \sqrt{7} \), which is what makes them like terms.
• By factoring \( \sqrt{7} \) out, we can transform the expression into \( \left( \frac{3}{4} - \frac{4}{3} \right) \sqrt{7} \).

This skill of combining like terms is especially useful in simplifying expressions and solving equations, as it helps to consolidate components and focus primarily on their coefficients.

Common Denominator

Finding a common denominator is a necessary step when you need to add or subtract fractions with different denominators. In this problem, the expression \( \frac{3}{4} - \frac{4}{3} \) requires a common denominator to be combined properly.

• The least common denominator of the fractions \( \frac{3}{4} \) and \( \frac{4}{3} \) is 12.
• To adjust the fractions to have this common denominator, we convert \( \frac{3}{4} \) into \( \frac{9}{12} \) and \( \frac{4}{3} \) into \( \frac{16}{12} \).

Once both fractions have the same denominator, it becomes straightforward to perform operations such as addition or subtraction on them. Identifying and using a common denominator simplifies the process and ensures accuracy in calculations.

Fraction Subtraction

Subtracting fractions involves a few straightforward steps. Once you've established a common denominator, such as 12 here, you can subtract the fractions easily. In our example, we need to subtract \( \frac{16}{12} \) from \( \frac{9}{12} \).

• Align the fractions for subtraction: \( \frac{9}{12} - \frac{16}{12} \).
• Subtract the numerators while keeping the common denominator: the calculation \( 9 - 16 = -7 \) results in the fraction \( \frac{-7}{12} \).

Thus, the expression simplifies to \( -\frac{7}{12} \sqrt{7} \), embodying effective fraction subtraction. This approach keeps the process organized and ensures the accurate simplification of algebraic expressions.