Problem 13 - College Algebra

Question

Simplify each expression. $$\sqrt{\frac{81}{5}}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{9\sqrt{5}}{5} \).

Step 1: Simplify the Radical Expression

To simplify the expression \( \sqrt{\frac{81}{5}} \), first evaluate the square root of the numerator separately. The numerator is 81, which can be written as \( 9^2 \). Therefore, \( \sqrt{81} = 9 \).

Step 2: Apply the Square Root to the Denominator

The square root of the denominator is \( \sqrt{5} \). Since 5 is not a perfect square, \( \sqrt{5} \) remains as it is.

Step 3: Combine and Simplify the Expression

Now, combine the results from the numerator and the denominator. The expression \( \sqrt{\frac{81}{5}} = \frac{\sqrt{81}}{\sqrt{5}} = \frac{9}{\sqrt{5}} \).

Step 4: Rationalize the Denominator

To complete the simplification, rationalize the denominator by multiplying both the numerator and the denominator by \( \sqrt{5} \): \[ \frac{9}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{9\sqrt{5}}{5} \]

Step 5: Conclusion: Simplified Expression

The simplified expression for \( \sqrt{\frac{81}{5}} \) is \( \frac{9\sqrt{5}}{5} \).

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics, representing a value that, when multiplied by itself, gives the original number. For instance, the square root of 81 is 9 because \( 9 \times 9 = 81 \). The symbol used to denote a square root is \( \sqrt{} \).
In our example, we look at \( \sqrt{\frac{81}{5}} \). The first step is to calculate the square root of the numerator, 81. Since 81 is a perfect square, this calculation is simple. Understanding square roots helps in breaking down complex radicals into more manageable forms, especially when dealing with fractions or algebraic expressions.
Always remember that the square root operation can only yield non-negative results for real numbers, which is why we don't consider negative roots unless specified.

Rationalizing the Denominator

Rationalizing the denominator involves removing any square roots from the denominator of a fraction. Typically, mathematicians prefer fractions with rational numbers (whole numbers or those that can be expressed as a fraction of two integers) in the denominator.
In the expression \( \frac{9}{\sqrt{5}} \), the denominator \( \sqrt{5} \) is not rational. To eliminate the square root, we multiply both the numerator and the denominator by \( \sqrt{5} \).

• The numerator becomes \( 9\sqrt{5} \).
• The denominator becomes \( \sqrt{5} \times \sqrt{5} = 5 \).

This process yields \( \frac{9\sqrt{5}}{5} \), a fraction now with a rational denominator.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations. In simplifying radical expressions like \( \sqrt{\frac{81}{5}} \), understanding algebraic manipulation is key.
We initially handle the expression by considering both the numerator and the denominator separately, treating \( 81 \) and \( 5 \) as individual algebraic parts. We find the square root of 81 separately to simplify the process. By rationalizing the denominator, we further simplify the algebraic expression into a final manageable form, \( \frac{9\sqrt{5}}{5} \).
Skills in simplifying and manipulating such expressions are vital, especially when handling more complex formulas in algebra.

Perfect Squares

Perfect squares are integers that are the square of other integers. For example, 1, 4, 9, 16, 25, 36, and so on are perfect squares. These numbers are important because their square roots are whole numbers, simplifying many mathematical processes.
In the context of \( \sqrt{\frac{81}{5}} \), recognizing that 81 is a perfect square allows us to easily compute its square root, which is 9. This step reduces the complexity of the radical expression significantly.
Being able to identify perfect squares quickly is a useful skill that simplifies many mathematical problems, leading to faster and more efficient solutions.