Problem 57 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-80: Simplify the expression, and rationalize the denominator when appropriate. $$ \sqrt{81} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt{81} \) is 9.

Step 1: Understanding the Problem

We are given the expression \( \sqrt{81} \) and are asked to simplify it. Simplifying a square root means finding the number which, when multiplied by itself, gives the original number.

Step 2: Identify the Perfect Square

The number inside the square root, 81, can be expressed as a perfect square. A perfect square is a number that can be expressed as the square of an integer. We check common perfect squares to identify that \( 81 = 9^2 \).

Step 3: Simplifying the Square Root

Since \( 81 = 9^2 \), we can use the property of square roots: \( \sqrt{a^2} = a \) for \( a \geq 0 \). Applying this property, \( \sqrt{81} = \sqrt{9^2} = 9 \).

Step 4: Conclusion

The simplified form of the expression \( \sqrt{81} \) is 9. Since the problem does not involve any fractions, and thus no denominator exists, rationalization is not needed in this case.

Key Concepts

Perfect Squares

A perfect square is a number that results from squaring an integer. For example, when you multiply the integer 9 by itself (9 x 9), you get 81, making 81 a perfect square. Recognizing perfect squares is key when simplifying square roots because it allows you to easily find the square root without performing complex calculations. Some common perfect squares include:

• 1 = 1 x 1
• 4 = 2 x 2
• 9 = 3 x 3
• 16 = 4 x 4
• 25 = 5 x 5
• ... and so on

When you encounter a square root, check if the number is a perfect square. If it is, like in the case of 81, you can directly conclude that the square root is an integer (9 in this case). This recognition makes solving these problems much more straightforward.

Square Root Properties

Square roots have fundamental properties that help us simplify expressions. Here are some key properties:

• The square root of a product \( ab \) is the product of the square roots: \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \) if both a and b are non-negative.
• The square root of a square returns the original number: \( \sqrt{a^2} = a \) for \( a \geq 0 \).
• For non-negative numbers, \( \sqrt{a} \cdot \sqrt{a} = a \).

Let's break it down with an example. If you have \( \sqrt{81} \), you know that \( 81 = 9^2 \). Using the property \( \sqrt{a^2} = a \), we find that \( \sqrt{81} = 9 \). This property allows us to simplify square roots quickly when dealing with perfect squares.

Rationalization of Denominators

Rationalization is the process of eliminating square roots from the denominator of a fraction. In math, it's common practice to express numbers such that all radical terms are in the numerator. Although not needed in the given example, it's still a crucial concept. Here is how you do it:1. **Identify the Square Root in the Denominator:** If you have \( \frac{a}{\sqrt{b}} \), where \( b \) is not a perfect square, the denominator is not already rational.2. **Multiply by the Conjugate:** To rationalize, multiply the numerator and denominator by \( \sqrt{b} \). This action will cancel out the square root in the denominator because \( \sqrt{b} \cdot \sqrt{b} = b \).3. **Simplify the Result:** Now your expression should have a rational denominator, converting it to a simpler form.For example, if you start with \( \frac{3}{\sqrt{2}} \), multiplying the numerator and denominator by \( \sqrt{2} \) gives \( \frac{3\sqrt{2}}{2} \). This process keeps expressions neat and follows mathematical conventions.