Problem 61 - Algebra A Combined Function

Question

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt{14}}{\sqrt{7}}\)

Step-by-Step Solution

Answer

\( \frac{\sqrt{14}}{\sqrt{7}} = \sqrt{2} \).

Step 1: Identify the Quotient Rule

The quotient rule for square roots states that you can divide the square roots in a fraction as follows: \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). Here, we have \( \frac{\sqrt{14}}{\sqrt{7}} \).

Step 2: Apply the Quotient Rule

Using the quotient rule, rewrite the expression as a single square root: \( \sqrt{\frac{14}{7}} \).

Step 3: Simplify the Fraction Inside the Square Root

Divide \(14\) by \(7\), which results in \(2\). This gives us \( \sqrt{2} \).

Step 4: Final Simplified Result

After applying the quotient rule and simplifying, the expression \( \frac{\sqrt{14}}{\sqrt{7}} \) simplifies to \( \sqrt{2} \).

Key Concepts

Square Roots Simplification

When dealing with square roots, it's often necessary to simplify them to make calculations easier and more understandable. The concept of square roots involves finding a number which, when multiplied by itself, gives the original number. For example, the square root of 4 is 2, because 2 multiplied by itself equals 4.

In simplifying square roots within fraction form, the quotient rule is particularly helpful. It allows us to combine square roots under a single radical, simplifying the process to a more straightforward calculation. By using the quotient rule in our example, we transformed \( \frac{\sqrt{14}}{\sqrt{7}} \) into \( \sqrt{\frac{14}{7}} \), leading us to a simpler and more easily managed expression.

Simplifying square roots can help streamline more complex algebraic expressions and solve equations efficiently, making it an essential skill in algebra.

Simplifying Radicals

Radicals can seem complex, but simplifying them follows straightforward rules. The goal is to rewrite a radical expression to its simplest form. Often this involves dividing radicals, factoring numbers to find perfect squares, or combining like terms inside a radical.

Here's how you simplify a radical expression:

• Identify any perfect square factors in the number under the radical.
• Extract the square root of these factors.
• Rewrite the expression with the extracted factors outside the radical.

In our example of \( \sqrt{14} \) and \( \sqrt{7} \), when applying the quotient rule, it became \( \sqrt{\frac{14}{7}} \). Simplifying the fraction \( \frac{14}{7} \), we found 2, which does not have further factors to simplify, thus remaining as \( \sqrt{2} \).

Simplifying radicals helps clarify expressions and makes them easier to work with, as we transformed our fraction to obtain a simple radical like \( \sqrt{2} \).

Fractional Division in Algebra

Fractional division is a common operation in algebra, often involving division of one fraction by another or dividing terms within a fraction. The quotient rule for square roots specifically applies when dividing two square roots, transforming it into a single square root, as seen in our example from the exercise with \( \frac{\sqrt{14}}{\sqrt{7}} \).

To divide fractions in algebra, remember these steps:

• Analyze if the division can be simplified using basic arithmetic operations, such as factoring or cancelling common terms.
• Utilize rules like the quotient rule, which helps condense square roots into manageable expressions.
• Perform any arithmetic operations inside the resultant expression to further simplify.

By using these methods, fractional division becomes a manageable and streamlined process in algebra, turning more complex expressions into simpler forms like \( \sqrt{2} \), helping in both basic arithmetic and more advanced calculations.