Problem 39 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ \sqrt{\frac{28}{x^{2}}}+\sqrt{\frac{7}{4 x^{2}}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{5\sqrt{7}}{2x} \).

Step 1: Simplify the First Expression

The first term is \( \sqrt{\frac{28}{x^2}} \). We can simplify under the square root by breaking it into two parts: \( \sqrt{28} \) and \( \sqrt{\frac{1}{x^2}} \). Since \( \sqrt{\frac{1}{x^2}} = \frac{1}{x} \), the entire term becomes \( \frac{\sqrt{28}}{x} \). Further simplify \( \sqrt{28} \) to \( 2\sqrt{7} \) as 28 can be factored into \( 4 \times 7 \). Thus, the simplified form is \( \frac{2\sqrt{7}}{x} \).

Step 2: Simplify the Second Expression

The second term is \( \sqrt{\frac{7}{4x^2}} \). This can be broken into \( \sqrt{7} \) and \( \sqrt{\frac{1}{4x^2}} \). Since \( \sqrt{\frac{1}{4x^2}} = \frac{1}{2x} \), the entire term simplifies to \( \frac{\sqrt{7}}{2x} \).

Step 3: Add the Two Simplified Expressions

Now, add the simplified terms: \( \frac{2\sqrt{7}}{x} + \frac{\sqrt{7}}{2x} \). Notice that they have a common denominator of \( x \). To add them, adjust the denominators to be the same. The common denominator should be \( 2x \)\ in this case: \( \frac{4\sqrt{7}}{2x} + \frac{\sqrt{7}}{2x} = \frac{(4\sqrt{7} + \sqrt{7})}{2x} \). Combine like terms in the numerator to get \( \frac{5\sqrt{7}}{2x} \).

Step 4: Final Result

The expression \( \sqrt{\frac{28}{x^2}} + \sqrt{\frac{7}{4x^2}} \) simplifies to \( \frac{5\sqrt{7}}{2x} \).

Key Concepts

Addition and Subtraction of Radicals

Adding and subtracting radicals can seem tricky at first, but it's quite similar to working with algebraic expressions. The key is to treat radicals like variables. You can only combine radicals that have the same radicand, which is the number under the square root. For example, in the expression \( \sqrt{28} + \sqrt{7} \), you can't combine these because the radicands 7 and 28 are different.

When the radicals are simplified to have the same radicand, you can easily add or subtract them just like terms in algebra, keeping the shared radical part and adding or subtracting the coefficients (the number in front of the square root).

• The expression \( 2\sqrt{7} + 3\sqrt{7} \) becomes \( 5\sqrt{7} \) because the radicands are the same (7).
• Similarly, \( \sqrt{3} - 2\sqrt{3} \) simplifies to \( -\sqrt{3} \).

Practicing this skill will make it easier to solve complex expressions involving radicals.

Real Numbers

Understanding real numbers is central to simplifying radical expressions. Real numbers include all the numbers on a continuous line of numbers. They encompass:

• Rational numbers, like \( 1/2 \) and \(-4\), which can be expressed as fractions.
• Irrational numbers, such as \( \pi \) and \( \sqrt{2} \), which can't be neatly written as fractions.

Radicals often result in irrational numbers, which means they don't create clean decimal equivalents and can't fully be written as a fraction.

When combining radicals, understanding that you're often dealing with irrational numbers helps you recognize why precision in simplification is necessary to achieve accurate results.

Common Denominator

When adding fractions, finding a common denominator is crucial. The same rule applies when adding or subtracting coefficients of radicals that include fractions. For instance, when you have expressions like \( \frac{2\sqrt{7}}{x} + \frac{\sqrt{7}}{2x} \), the denominators differ, and you must find a common denominator to combine them.

In this case, the least common denominator is \( 2x \), making it easier to sum the expressions. Here's how you achieve that:

• Multiply the first expression \( \frac{2\sqrt{7}}{x} \) by \( \frac{2}{2} \) to have a denominator of \( 2x \), resulting in \( \frac{4\sqrt{7}}{2x} \).
• The common denominator ensures that the fractions add smoothly, allowing you to compute: \( \frac{4\sqrt{7}}{2x} + \frac{\sqrt{7}}{2x} \).

The final combined expression, \( \frac{5\sqrt{7}}{2x} \), demonstrates the power of a common denominator in simplifying radical expressions.

Square Roots

Square roots open a fascinating world in mathematics. A square root provides a number which, when multiplied by itself, results in the original number. For instance, \( \sqrt{9} = 3 \), as \( 3 \times 3 = 9 \).

To simplify a square root like \( \sqrt{28} \), factor it into prime components such as \( 4 \times 7 \). Since \( 4 \) is a perfect square, pull it out of the square root to get \( 2\sqrt{7} \):

• Break down the radicand into factors.
• Identify any perfect squares, which can be taken outside of the square root.
• Simplify the expression, keeping the square root of non-perfect squares inside.

This method is crucial for clearing complex radical expressions, allowing for appropriate additions or subtractions.