Problem 16 - Intermediate Algebra

Question

Add or subtract. $$ \frac{3 x \sqrt{7}}{5}+\sqrt{\frac{7 x^{2}}{100}} $$

Step-by-Step Solution

Answer

The answer is \( \frac{7x\sqrt{7}}{10} \).

Step 1: Simplifying the Second Expression

First, simplify the second expression \( \sqrt{\frac{7 x^2}{100}} \). This can be rewritten as \( \sqrt{\frac{7}{100} \cdot x^2} \), which is equal to \( \frac{\sqrt{7} \cdot x}{10} \) after taking the square root of the numerator and the denominator.

Step 2: Rewriting to a Common Denominator

Now, find a common denominator for both terms. The first term is \( \frac{3x\sqrt{7}}{5} \) and the second term is \( \frac{x\sqrt{7}}{10} \). The least common denominator (LCD) of 5 and 10 is 10. So, rewrite the first term with this denominator: \( \frac{6x\sqrt{7}}{10} \) since \( \frac{3x\sqrt{7}}{5} = \frac{6x\sqrt{7}}{10} \).

Step 3: Adding the Fractions

Now that both terms have the same denominator, add them together: \( \frac{6x\sqrt{7}}{10} + \frac{x\sqrt{7}}{10} \). Since they have the same denominator, you can simply add the numerators: \( \frac{(6x\sqrt{7} + x\sqrt{7})}{10} \).

Step 4: Combine Like Terms

Combine the like terms in the numerator: \( 6x\sqrt{7} + x\sqrt{7} = 7x\sqrt{7} \).

Step 5: Simplified Expression

The expression now is \( \frac{7x\sqrt{7}}{10} \). This is the simplified form of the original expression after performing the addition.

Key Concepts

Least Common Denominator

When working with fractions, finding a common denominator is crucial for addition or subtraction. This common denominator is known as the least common denominator (LCD). It is the smallest number that both denominators can divide into without leaving a remainder. Finding the LCD allows us to align fractions for easy arithmetic operations.

• To find the LCD, identify the denominators of the fractions you are dealing with.
• Determine the least common multiple (LCM) of these denominators.
• Rewrite each fraction with the LCD as their new common denominator.

In the given problem, the denominators of the fractions are 5 and 10. The LCD of 5 and 10 is 10. Therefore, we modify the first fraction (\( \frac{3x\sqrt{7}}{5} \)) using a factor of 2 to convert 5 to 10, resulting in \( \frac{6x\sqrt{7}}{10} \). Having a common denominator simplifies the addition of the two fractions.

Simplifying Radicals

Radicals can initially appear complex, but simplifying them often makes the problem easier to handle. Simplifying a radical involves rewriting it in its simplest form.

• Break down the expression under the radical into its simplest components.
• Separate the factors beneath the radical sign for ease of simplification.
• Extract any perfect squares from the radical, if applicable.

For example, consider \( \sqrt{\frac{7x^2}{100}} \). Break it down as \( \sqrt{\frac{7}{100} \cdot x^2} \). Simplify by taking the square root of each factor: \( \frac{\sqrt{7} \cdot x}{10} \). This approach transforms the radical into a more manageable expression, making it easier to integrate with other components in an equation.

Adding Fractions

Adding fractions involves combining fractions with a common denominator. If the denominators differ, adjust them to a common denominator first. This process is straightforward once the denominators match.

• Ensure all fractions have the same denominator.
• Add the numerators while keeping the denominator constant.
• Simplify the resulting fraction if possible.

In this exercise, after equating the denominators to 10, we add \( \frac{6x\sqrt{7}}{10} \) to \( \frac{x\sqrt{7}}{10} \). Simply sum the numerators: \( 6x\sqrt{7} + x\sqrt{7} \), and place the sum over the common denominator. The resulting fraction, \( \frac{7x\sqrt{7}}{10} \), is a simplified version where no further reduction is needed.

Algebraic Expressions

Understanding algebraic expressions is essential for handling terms like \( 3x\sqrt{7} \) that involve variables and constants. These expressions comprise components such as terms, coefficients, and variables.

• Each part of an algebraic expression can represent a specific mathematical value or function.
• Combining like terms simplifies the expression, typically by addition or subtraction.
• An expression may include variables (e.g., x), coefficients (numerical values before variables), and radicals (e.g., \( \sqrt{7} \)).

In the context of the problem, \( 6x\sqrt{7} + x\sqrt{7} \) illustrates combining like terms. Here, both terms share \( x\sqrt{7} \), hence the coefficients (6 and 1) are added to yield \( 7x\sqrt{7} \). Simplifying algebraic expressions through this method makes further mathematical operations easier to perform.