Problem 11 - Intermediate Algebra

Question

For Problems \(1-20\), use the distributive property to help simplify each of the following. For example, $$ \begin{aligned} 3 \sqrt{8}-\sqrt{32} &=3 \sqrt{4} \sqrt{2}-\sqrt{16} \sqrt{2} \\ &=3(2) \sqrt{2}-4 \sqrt{2} \\ &=6 \sqrt{2}-4 \sqrt{2} \\ &=(6-4) \sqrt{2}=2 \sqrt{2} \end{aligned} $$ $$ \frac{3}{4} \sqrt{7}-\frac{2}{3} \sqrt{28} $$

Step-by-Step Solution

Answer

\( \frac{-7}{12} \sqrt{7} \)

Step 1: Factor the Radicals

Begin by factoring the numbers under the square roots to see if they can be expressed in terms of another square root multiplied by a square number. For example, 28 can be factored to 4 and 7, giving us: \[ \frac{3}{4}\sqrt{7} - \frac{2}{3}\sqrt{4 \times 7}.\]

Step 2: Simplify the Radicals

Since \(\sqrt{4} = 2\), rewrite \(\sqrt{28}\) as \(2\sqrt{7}\). The expression becomes: \[ \frac{3}{4}\sqrt{7} - \frac{2}{3} \times 2\sqrt{7} = \frac{3}{4}\sqrt{7} - \frac{4}{3}\sqrt{7}.\]

Step 3: Factor Out Common Radicals

Factor out the common term \(\sqrt{7}\) from both terms: \[ (\frac{3}{4} - \frac{4}{3}) \sqrt{7}.\]

Step 4: Simplify the Coefficients

To combine the coefficients, first find a common denominator. The least common multiple of 4 and 3 is 12. Rewrite the coefficients with this common denominator: \[ \frac{3}{4} = \frac{9}{12} \text{ and } \frac{4}{3} = \frac{16}{12}.\]

Step 5: Perform the Subtraction

Subtract the fractions: \[ \frac{9}{12} - \frac{16}{12} = \frac{-7}{12}.\]

Step 6: Write the Final Expression

The final simplified expression is: \[ \frac{-7}{12} \sqrt{7}.\]

Key Concepts

Simplifying Radicals

In mathematics, simplifying radicals means to express a radical expression in its simplest form. Think of it as cleaning up the expression so it looks tidy. A radical like \(\sqrt{28}\) can be simplified by identifying perfect square factors.

• For example, \(28\) can be broken down into \(4 \times 7\).
• Since \(\sqrt{4} = 2\), you can simplify \(\sqrt{28}\) to \(2\sqrt{7}\).

This process makes it easier to work with the radical because you're handling smaller numbers or simpler forms. Simple radicals are crucial when you intend to add or subtract radicals because only like radicals can be combined. By reducing each radical to its simplest form, operations with radicals become more streamlined and manageable.

Factoring Radicals

Factoring radicals is all about splitting the number under the radical sign into two or more factors, especially when one of them is a perfect square. This method is considered a cornerstone for simplifying radicals.Why Factor Radicals?

• To simplify a radical expression, making it easier to manipulate mathematically.
• To allow combinations such as addition or subtraction with other like radicals.

For instance, take \(\sqrt{28}\). We know that \(28\) can be expressed as \(4 \times 7\). Here, \(4\) is a perfect square, which simplifies to \(2\). By factoring radicals, we achieve \(2\sqrt{7}\). This simplification can aid in further steps such as addition, subtraction, or even solving equations involving radicals.

Common Denominator

The concept of a common denominator is a key part of operations involving fractions. It allows us to combine or compare fractions more easily by aligning them to a common scale. To find a common denominator:

• Determine the least common multiple (LCM) of the denominators involved.
• Rewrite each fraction using this common denominator.

In the example, we had fractions with denominators \(4\) and \(3\). Their LCM is \(12\). So, we rewrite \(\frac{3}{4}\) as \(\frac{9}{12}\) and \(\frac{4}{3}\) as \(\frac{16}{12}\). Now, both fractions are on the same scale, making subtraction straightforward: \(\frac{9}{12} - \frac{16}{12} = \frac{-7}{12}\). This technique is essential not just in arithmetic, but in algebraic manipulation involving fractions as well.