Problem 26 - Algebra and Trigonometry

Question

Use the quotient rule to simplify the expressions Assume that \(x>0\). $$ \sqrt{\frac{121}{9}} $$

Step-by-Step Solution

Answer

The simplified expression of \( \sqrt{\frac{121}{9}} \) is \( \frac{11}{3} \)

Step 1: Identify Numerator and Denominator

In this exercise, the numerator inside the square root is 121 and the denominator is 9

Step 2: Apply the Quotient Rule

The quotient rule states that the square root of a fraction is equal to the square root of the numerator divided by the square root of the denominator

Step 3: Take the Square Roots

Square root of 121 is 11 and square root of 9 is 3.

Step 4: Divide

Now, divide 11 by 3 to get the answer

Key Concepts

Square Roots

Square roots help us find a number that, when multiplied by itself, gives another specified number. The square root symbol \( \sqrt{\cdot} \) is used to represent this operation.

When working with square roots, especially with fractions, the quotient rule is a handy tool. This rule simplifies the process by allowing us to take separate square roots of both the numerator and the denominator.

For instance, consider the expression \( \sqrt{\frac{121}{9}} \). Instead of directly trying to find this square root as a whole, the quotient rule helps break it down:

• First, identify the numerator: 121.
• Next, identify the denominator: 9.

With the quotient rule, you now calculate \( \sqrt{121} \) and \( \sqrt{9} \) separately, making it simpler to handle.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form. The aim is to make problems easier to understand and solve. Let's explore how this fits into solving square roots of fractions.

Once you have calculated the square roots of the numerator and denominator separately, you then simplify the expression by dividing these two results. This final step is often the easiest.

In our example:

• The square root of 121 is 11.
• The square root of 9 is 3.

The expression thus simplifies to \( \frac{11}{3} \). It can't be simplified further, since 11 and 3 have no common factors other than 1.

Through this method, complex expressions can be neatly reduced to their simplest terms, easing both computation and understanding.

Fraction Division

Understanding how to divide fractions is key in many math problems. It becomes especially important when fractions are involved in roots and requires step-by-step simplification.

Once you've calculated the square roots of both parts of your fraction, the task becomes straightforward: you divide the numerator by the denominator. This operation appears frequently in algebra, often wrapped in problems involving roots.

For dealing with simple division of fractions, remember:

• Place the numerator (top number, here 11 after square root) over the denominator (bottom number, here 3 after square root).
• Carry out the division as you would with regular numbers.

By carrying out these steps, the result is a simplified value representing the original expression neatly as \( \frac{11}{3} \), remaining in fractional form if necessary for precision.