Problem 13 - Algebra A Combined Function

Question

Use the quotient rule to simplify. Assume that all variables represent positive real numbers. $$ \sqrt{\frac{6}{49}} $$

Step-by-Step Solution

Answer

\(\frac{\sqrt{6}}{7}\)

Step 1: Express the Square Root as a Fraction

The given expression is \( \sqrt{\frac{6}{49}} \). A square root of a fraction \( \frac{a}{b} \) can be expressed as \( \frac{\sqrt{a}}{\sqrt{b}} \). This step splits the square root of the fraction into the quotient of individual square roots.

Step 2: Simplify the Square Roots

Now simplify the expression \( \frac{\sqrt{6}}{\sqrt{49}} \). The square root of 49 is 7, so we have \( \frac{\sqrt{6}}{7} \). \( \sqrt{6} \) remains as it is because 6 is not a perfect square.

Key Concepts

Simplifying Square Roots

When working with expressions that involve square roots, simplifying them can become much easier by breaking down the structure. Square roots are often used to represent the principal square root, which is the non-negative value that, when multiplied by itself, gives the original number. To simplify square roots, we look for perfect squares that can be factored out of the number under the square root.

For example, consider the number 18. It can be expressed as the product of its prime factors: 18 = 2 × 3 × 3. Here, 3 × 3 is a perfect square, which is 9, so the square root of 18 can be simplified as follows:

• Express 18 as the product of perfect squares: \( \sqrt{18} = \sqrt{2 \times 9} \)
• The square root of 9 is 3, so we can write: \( \sqrt{18} = 3 \sqrt{2} \)

This method significantly simplifies calculations. When simplifying square roots, always check to see if the number can be broken down into smaller multiples that include a perfect square.

Fractional Expressions

Fractional expressions often appear in algebra and require careful handling to maintain their balance and integrity. A fractional expression generally involves a numerator and a denominator, separated by a division line. These expressions can include numbers, variables, or a combination of both. Simplifying fractional expressions mandates that you alter them to their simplest forms while preserving equivalent value.

It's essential to properly double-check for common factors in the numerator and the denominator. Removing these common factors simplifies the fraction. For example, take the expression \( \frac{8x}{12} \). Here’s how to simplify it:

• Determine the greatest common factor (GCF): The GCF of 8 and 12 is 4.
• Divide both the numerator and the denominator by their GCF: \( \frac{8x}{12} = \frac{8x \div 4}{12 \div 4} = \frac{2x}{3} \).

By reducing both parts by the GCF, the fraction \( \frac{8x}{12} \) becomes \( \frac{2x}{3} \), a more manageable expression. This same method is useful when dealing with expressions that have variables in addition to constants.

Square Roots of Fractions

Dealing with square roots of fractions might initially appear daunting, but it becomes manageable once you understand and apply the quotient rule effectively. The key idea is that the square root of a fraction can be expressed as the quotient of the square roots of its numerator and its denominator. This rule is formally expressed as:

• For any fraction \( \frac{a}{b} \), where both \( a \) and \( b \) are non-negative, \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).

Applying this, let’s revisit the initial expression from the exercise: \( \sqrt{\frac{6}{49}} \). Using the quotient rule:

• Separate the square root of the fraction into the fractions of square roots: \( \frac{\sqrt{6}}{\sqrt{49}} \).
• Recognize that the square root of 49 is a perfect square: \( \sqrt{49} = 7 \).

Consequently, the expression simplifies further to \( \frac{\sqrt{6}}{7} \), where 6 remains under the radical as it cannot be simplified further. Breaking down square roots of fractions into more manageable parts helps simplify the process, making complex expressions much more approachable.