Problem 13 - Intermediate Algebra

Question

Use the quotient rule to simplify. See Examples 2 and 3 . $$ \sqrt{\frac{6}{49}} $$

Step-by-Step Solution

Answer

\( \sqrt{\frac{6}{49}} = \frac{\sqrt{6}}{7} \)

Step 1: Identify the Quotient Rule for Radicals

The quotient rule for radicals states that \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). We can apply this rule to simplify \( \sqrt{\frac{6}{49}} \).

Step 2: Apply the Quotient Rule

Use the quotient rule to express the square root of the fraction as a fraction of square roots. Therefore, \( \sqrt{\frac{6}{49}} = \frac{\sqrt{6}}{\sqrt{49}} \).

Step 3: Simplify the Denominator

The square root of 49 is 7 because \( 7^2 = 49 \). Hence, \( \frac{\sqrt{6}}{\sqrt{49}} = \frac{\sqrt{6}}{7} \).

Step 4: Finalize the Expression

The expression is already simplified as \( \frac{\sqrt{6}}{7} \). Since 6 is not a perfect square, \( \sqrt{6} \) cannot be simplified further.

Key Concepts

Simplifying Radicals

Simplifying radicals means breaking down expressions under a square root into their simplest forms. The process often involves identifying and factoring out perfect squares. However, if the number is not a perfect square, you might not be able to simplify further. For instance, in the exercise, after applying the quotient rule, we end with the square root of 6. Since 6 has no perfect square factors—other than 1—it cannot be simplified further.

• When simplifying a radical, always look for perfect square factors that can be removed from under the square root sign.
• For example, the square root of 36 simplifies to 6 since 36 is a perfect square.
• If there are no perfect squares, the expression is already in its simplest form.

Understanding these basics can help you manage radicals with greater confidence, especially when dealing with fractions.

Square Root

The square root is a mathematical operation that finds a number which, when multiplied by itself, results in the original number. Mathematically, the square root of a number \( x \) is written as \( \sqrt{x} \). For instance, \( \sqrt{49} \) equals 7 because \( 7 \times 7 = 49 \).

• If a number is a perfect square, its square root will be an integer. This simplifies calculations.
• Non-perfect squares result in square roots that are irrational numbers, often requiring approximate or exact fractional representations.
• It's important to remember the properties of square roots when using the quotient rule to break down more complex radicals.

Learning how to manage square roots, particularly within fractions, can greatly simplify algebraic expressions and problem-solving.

Fractions in Algebra

Fractions in algebra show the relationship between two numbers, one on top of the other, separated by a line. The number above is the numerator, and the one below is the denominator. In algebra, it's crucial to simplify these fractions whenever possible.

• The quotient rule helps in simplifying fractions under a square root by expressing the square root of a fraction as a fraction of square roots.
• In our example, the fraction \( \frac{6}{49} \) underwent this transformation, simplifying to \( \frac{\sqrt{6}}{7} \).
• By simplifying fractions, you make algebraic expressions easier to handle and understand.

Algebra dealing with fractions often includes finding common denominators or reducing fractions to their simplest form, best done after ensuring that any radicals are simplified too.